physics honors - cardinal spellman high school bruno...physics honors introduction to physics 1...

PHYSICS HONORS

Table of Contents

Ch. 1 Introduction to Physics…………………………………………………..…. .…….1 – 17

Ch. 2 Kinematics…………………………………………………………………… …….18 – 25

Ch. 3 Statics and Dynamics………………………………………………………. …….26 – 33

Ch. 4 Work and Thermodynamics……………………………………………….. …….34 – 47

Ch. 5 Fluid Mechanics…………………………………………………………….. …….48 – 52

Ch. 6 Mechanical Waves………………………………………………………….. …….53 – 59

Ch. 7 Light and Optics…………………………………………………………….. …….60 – 69

Ch. 8 Electricity……………………………………………………………………. …….70 – 78

Ch. 9 Magnetism…………………………………………………………………… …….79 – 85

Ch. 10 Quantum Physics…………………………………………………………… …….86 – 93

Appendix A: Average Heat Constants…………………………………….. ……….….94

Glossary………………………………………………………………………. .….95 – 108

Information provided may be influenced by the following sources:

Amsco Science Proficiency Review 2000

Amsco Physics: A Contemporary Approach 2000

Amsco Physical Science Work-Text 2001

Amsco Reviewing Physics 2007

Barron’s Let’s Prepare series 2002

Free High School Science Texts version 0

Glencoe NY Regents Review Series 2005

Prentice Hall Brief Review Series 2013 LeBel Physics: Systems and Applications 2002

Physics for K–12 available for free at Connexions <http://cnx.org/content/col10322/1.175>

Frank D and Nykamp DQ, “An introduction to vectors.” From Math Insight. http://mathinsight.org/vector_introduction

http://mathinsight.org/contributor/dfrank

http://mathinsight.org/contributor/dqnykamp

Physics Honors Introduction to Physics

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Chapter 1 – Introduction to Physics

Physics is often described as the study

of matter and energy. It deals with the fundamental laws of nature and many of their applications. These laws govern the behavior of all physical phenomena. Physics may be divided into two general areas: experimental and theoretical. Experimental physicists design and run careful investigations on a broad range of natural phenomena under conditions which are uncharacteristic to those of our everyday lives. Theoretical physicists propose and develop models and theories to mathematically explain the results of experimental observations. Science is both a body of knowledge and a way of knowing things. Through intellectual and social activities, human thinking is applied to discovering and explaining how the world works. Science originates when people ask questions. Scientists use scientific inquiry and skills to seek answers to questions about the world. This involves formulating hypotheses, designing and conducting experiments, collecting data, and analyzing the data to form conclusions. Scientists also rely on peer review by other scientists to confirm the validity of their results. At one time, “scientific” knowledge was just a collection of opinions and unrelated ideas attempting to explain observations. Today, scientists do more than debate whether or not a new opinion or idea seems to make sense. They develop explanations using observations as evidence. New information is combined with what people already know. Learning about the historical development of scientific concepts and about the individuals who have contributed to scientific knowledge helps people understand the thinking that has taken place. Scientific investigation involves, among other factors, the following process skills:

observing

inferring

predicting

modeling

Observations are events that are made using any of the senses or tools which extend the senses, such as thermometers, graduated

cylinders, balances, or rulers. The development of better tools increases the ability of scientists to observe the natural world. There are differences between qualitative and quantitative observations. A qualitative observation describes that situation and reaction in descriptive terms. For example, qualitative data involves the five senses and includes color and shape. A quantitative observation includes measurements. For example, describing something as a square would be qualitative, but measuring it and stating that it is 2 cm by 2 cm is quantitative. Conclusions or deductions based on observations are inferences. Inferences may be very subtle. An inference is an interpretation of an observation. It is a mental process that proposes causes, conclusions, or explanations for what has been observed. Inference may or may not be correct. Additional observations may make the inference more likely to be true. Inferences are not to be confused with assumptions or opinions. A good experiment keeps these to a minimum. An assumption is the belief that something is true. Assumptions also may be very subtle, and at first you may be unaware you are making them. Ideas people have that may or may not have any basis in fact are opinions. Opinions are often biased, or influenced by an assumption that may or may not be correct. Although everyone has opinions, which should be respected, a good way to avoid bias is to leave opinions out of data collection and analysis. A prediction of a future event is a type of inference. Misconceptions about environmental characteristics often result from incorrect inferences that become commonly believed. A misconception is a mistaken belief or a misunderstanding. A model is a representation of something that is often too difficult (or impossible) to display directly. While a model is justified with experimental proof, it is only accurate under limited situations. Physicists use models for a variety of purposes. They are a way of


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explaining or demonstrating what might be happening and to predict what will occur in new situations. Models can help physicists analyze a scenario and perform a calculation, or they can be used to represent a situation in the form of a computer simulation. Scientists group together similar observations and inferences to make the study of objects and events in the environment more meaningful or easier to understand. This grouping is called classification. Understanding the scientific view of the world is essential to personal, societal, and ethical decision making. To think scientifically, you must critically analyze events, explanations, and ideas.

1.1 Scientific Inquiry Questioning is at the heart of science. Progress in science depends on people who not only wonder how the world works but who also take the time to develop questions that can be tested and answered. Before investing time and resources on research, it is important to find out what others have already learned. Most research plans begin with a thorough library search. This search may include the use of electronic information retrieval (the Internet and library databases), a review of the literature (scientific journals) and feedback from the investigator’s peers. This background work is done so that the researcher has a thorough understanding of the major concepts being investigated and any similar investigations. Inquiry involves developing and presenting proposals, including formal hypotheses, to test explanations. A good hypothesis attempts to explain what has been observed in a way that can be tested. It is a tentative answer to a question. Experiments, series of trials or tests, cannot prove a hypothesis; they can only either support the hypothesis or fail to support it. A hypothesis can help determine the organization of an experiment as well as what data to collect and how to interpret those data. Testing a hypothesis is valuable even when the hypothesis is not supported by experimental results, since new information is gained in the process of testing any hypothesis.

Designing a way to test the hypothesis requires the following: - selecting, acquiring, and possibly building

apparatus - considering safety precautions - planning how to avoid bias - a dependent or responding variable - what

will be measured - independent or manipulated variables -

factors that might influence the dependent variable and affect the interpretation of the results

- a control – a reference point used as a standard of comparison

- a controlled experiment – one in which the possible variables have been carefully considered and regulated so the results are due only to the independent variable you are testing

There should only be one variable being tested at one time. Large sample sizes and multiple trials provide more accurate information and therefore less probability of error due to chance. After carefully considering how well the predicted result and the actual result of the experiment correspond, a conclusion can be made. Scientists often use statistical analysis techniques to find the likelihood that their results were produced by chance. If the results differ only slightly, errors in measurement, genetic differences among the test organisms, or chance may be the reason. Once a significant pattern or relationship has been discovered as a result of data analysis, a scientist next tries to explain why these results were obtained. Scientific inquiry requires the ability to develop a written report for public scrutiny. Scientists report their findings in scientific journals or during presentations at professional meetings. Reports: - describe the hypothesis - include a literature review of previous studies - explain the experiment performed and its

data - provide the scientist’s conclusion and

suggestions for further study

A peer review, in which several scientists examine the details of an experiment, is an important part of the scientific process.


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Scientists are expected to question explanations proposed by other scientists, analyze the experimental procedures, examine the evidence, identify faulty reasoning, point out statements that go beyond the evidence, and suggest alternative explanations for the same observations. Peer review is one of the systems of checks and balances in science. One assumption of science is that other individuals could arrive at the same explanation if they had access to similar evidence. Experiments that cannot be repeated exactly with the same results have little worth. Evidence is a collection of facts offered to support the idea that something is true. Scientists accept evidence when it is supported by many facts. Until they have a large collection of evidence to support their thinking, scientists must remain neutral. Well-accepted scientific theories are supported by many different scientific investigations, often involving the contributions of individuals from different disciplines. A theory is an explanation for patterns in nature that is supported by scientific evidence and verified multiple times by various groups of researchers. Some theories include models to help visualize phenomena, whereas others do not. A law uses concise language to describe a generalized pattern in nature that is supported by scientific evidence and repeated experiments. Often, a law can be expressed in the form of a single mathematical equation. Laws and theories are similar in that they are both scientific statements that result from a tested hypothesis and are supported by scientific evidence. The biggest difference between a law and a theory is that a law describes a single action; a theory explains an entire group of related phenomena. Less broadly applicable statements are usually called principles (such as Pascal’s principle, which is applicable only in fluids), but the distinction between laws and principles often is not carefully made.

Laboratory investigations are sometimes considered an exciting part of a course. However, they may involve potentially

dangerous activities or materials. As a result, careful attention to safety procedures is critical. Read all of the directions for an investigation before you start to work. If you are unsure about any part of the lab procedures, check with your instructor.

Do not eat or drink in the laboratory.

Never inhale or taste any of the chemicals you are using in a laboratory.

If you spill a chemical or get any on your skin, wash it off immediately. Report the incident to your instructor.

Tell your instructor about any personal injury no matter how minor it may seem.

Tie back long hair, and keep loose clothing away from laboratory equipment, chemicals, and sources of heat and fire.

Do not use glassware that has cracks or large chips. Tell your instructor about the damage and get a replacement.

Use laboratory apparatus as it is intended to be used.

Do not use electrical equipment around water. If electrical cords seem to have exposed wires or if you get a shock handling electrical equipment, notify your instructor immediately. Do not attempt to disconnect the equipment yourself.

Turn off the water after you are done with it. Disconnect any electrical devices.

Clean your work area by returning materials to their appropriate places, washing and drying glassware according to your instructor’s instructions, and wiping off the lab surface.

Wash your hands thoroughly!

1.2 Graphing

Many times an investigation will involve finding out how changing one quantity affects the value of another. The quantity that is intentionally changed is called the independent variable. The quantity that changes due to the variation in the independent variable is called the dependent variable. Data generally refers to the results of trials, or tests, completed during experiments. It can be organized into diagrams, tables, charts, graphs, equations, and matrices. Scientists must then be able to interpret the organized data and make


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y

x

inferences, predictions, and conclusions based on those data. A data table is an important initial stage in making sense of the information collected during an experiment. When constructing a data table, the following is important: - Title the table so that it relates the

independent variable to the dependent variable.

- Column headings must include the dependent and independent variables as well as the units of measurement for each. Columns may also be needed for trial or setup numbers or other information.

- The independent variable is typically recorded first and in increasing order. The dependent variable is recorded to correspond with the independent variable.

Data collected in an experiment is often represented in graphical form. A graph makes it easier to determine whether there is a trend or pattern in the data. A trend is a pattern of observations that are occurring in a particular direction. Patterns and trends in data help us make more accurate predictions. There are four basic types of graphs:

pie or circle graph

bar graph

histogram

line graph

By convention, the independent variable is graphed on the x- or horizontal axis while the dependent variable is graphed on the y- or vertical axis. The axes are labeled with the quantities and their units are given in parenthesis. An appropriate linear scale, without any breaks, that accommodates the range of data is determined for each axis. It is not necessary to label every grid line nor is it necessary to use the same scale for each axis. The graph should be titled as the dependent variable versus the independent variable usually using the same title as the data table.

After the data points are plotted, a smooth line of best fit is drawn. The best-fit line or best-fit curve is a straight or curved line which approximates the relationship among a set of data points. This line usually does not pass through all measured points. Sometimes you

need information about a value that you have not determined during the experiment. Estimating a value between known data points is called interpolation and is a form of deductive reasoning. Sometimes the line of best fit is extrapolated. Extrapolation means extending the line beyond the region in which data was taken and is a form of inductive reasoning. This is important because the point where the extended line intersects the horizontal or vertical axis has physical significance. Extrapolation must be used with caution because you cannot be sure that the relationship between the variables remains the same for all conditions.

The slope of a graphed line often has a physical meaning. On an x-y coordinate system, the slope of a line is defined as the ratio for any two points on the line. In determining the slope of a graphed line, points directly from the data table can only be used if those points lie on the line of best fit. A horizontal line has a slope of zero. If a line is nearly horizontal, its slope has a small value. If a line slants steeply, its slope has a large value. A line that slopes downward to the right has a negative slope. The following formula is used to determine the slope or rate of change between the variables:

xΔ

yΔ

change horizontal

change vertical slope

Some of the common relationships that exist between measured quantities measured are revealed by the shapes of graphs:

Two quantities are directly proportional if an increase in one causes an increase in the other. The slope is a non-zero constant and the graph is a diagonal line moving up and to the right. The relationship is represented by the equation y = mx + b where m is the slope of the line and b is the y-intercept.


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Two quantities are inversely proportional if an increase in one causes a decrease in the other. The graph is a diagonal line downward and to the right or curved line moving downward and to the right. The relationship is represented by the equation y = m(1/x) + b.

Two quantities have a constant proportion if an increase in one causes no change in the other. The graph is a horizontal line represented by the equation y = b where the letter b is used to represent any constant value.

Two quantities have a direct squared relationship if an increase in the x-variable causes a squared increase in the y-variable. The graph is a curved line moving up and to the right. The relationship is represented by the equation y = mx2 + b.

Two quantities have a root relationship if an increase in one causes a squared decrease in the other. The graph is a curved line moving downward and to the right. The relationship is represented by

the equation y = mx + b but can also be expressed as y2 = mx + b.

Many changes in the environment occur in some orderly fashion in which the events constantly repeat called a cyclic change.

1.3 Making Measurements Physics is based on observations and measurements of the physical world. Scientists have developed tools for measurement and adopted standard conventions for describing natural phenomena.

A unit is a standard quantity with which other similar quantities can be compared. All measurements must be made with respect to some standard quantity. The official system of units used to measure matter is called SI which stands for System International and is based on the metric system. The SI system provides standardized units for scientific measurements. All measured quantities can be expressed in terms of the seven fundamental units listed in table 1-1.

Basic Unit Basic Quantity

meter (m) length

kilogram (kg) mass

second (s) time

ampere (A) current

kelvin (K) temperature

mole (mol) amount of substance

candela (cd) luminous intensity

Table 1-1: Fundamental units

y

x

y

x

y

x

y

x

y

x

y

x


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Derived units are combinations of two or more of the fundamental units and are used to simplify notation. Many times, in order to simplify notation, these derived units are named for a scientist who had made significant contributions in the field; some derived units are listed in table 1-2.

Quantity Measured Derived Unit

Acceleration m/s2

Area m2

Density kg/m3

Energy kgm2/s

2

Force kgm/s2

Power kgm2/s

3

Pressure kg/ms2

Speed m/s

Volume m3

SI prefixes are combined with SI base units to form new units that are larger or smaller than the base units by a multiple of 10. The symbol for the new unit consists of the symbol for the prefix followed by the symbol for the base unit. For example, 1000 meters can be expressed as 1 kilometer or 1 km, and 0.01 meter can be expressed as 1 centimeter or 1 cm. The table 1-3 lists some SI prefixes, the most commonly used prefixes are found in bold type.

Larger than base Smaller than base

P peta-

1015 d

deci- 10

–1

T tera-

1012

c

centi- 10

–2

G giga-

109

m milli-

10–3

M mega-

106

micro-

10–6

k kilo-

103

n nano-

10–9

h hecto-

102

p pico-

10–12

da deka-

101

f femto-

10–15

Analyzing units can help in solving

problems. The units on the left side of an equation must always be equivalent to the units on the right side of the equation. Quantities can

be added or subtracted only if they have the same units. Solving physics problems is a logical and creative task. There are certain practices that help this creative process. First, a careful reading of the problem is necessary to fully grasp the question being posed and the information being given. It is often useful to separately write out all of the given and required information. It is also good practice to make a sketch of the problem and visualize the physics taking place. A correct mental picture of the problem takes you a long way toward a correct solution. For the mathematical solution you need to identify and solve the appropriate equations. Finally, you should check and explore your result to be sure that the answer makes sense in the context of the problem. When a quantity is broken down in terms of the fundamental units we call this breakdown its dimension. The dimension represents the type of the quantity or property. Many equations in science, especially in physics, these equations must be dimensionally consistent. It is extremely useful to perform a dimensional analysis on any equation about which you are unsure. If the dimensional consistency is not there, it cannot be a correct equation. The rules are simple:

Two quantities can only be added or subtracted if they are of the same dimension.

Two quantities can only be equal if they are of the same dimension.

For example, it is perfectly valid to write 12 inches = 1 foot because both represent lengths even though their units are different. This will lead to a conversion factor that helps when solving mathematical equations. However, it is not valid to write x inches = y seconds because they have different dimensions, one represents length and the other time. Even though we predominantly use basic SI units it will often be necessary to convert between SI and other units. A conversion can be accomplished using a conversion factor that is constructed by knowing how much of a quantity in one unit equals that same quantity in another unit. A conversion factor is a ratio of equal quantities written such that, when multiplied by a

Table 1-2: Derived units

Table 1-3: Metric prefixes


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1.50 m

1.33 m

1.50 m

quantity, the undesired unit algebraically cancels leaving only the desired unit. The following example illustrates the use of conversion factors. Question A cardboard box measures 1.50 m x 1.50 m x 1.33 m. Determine the volume of clothes that you can pack into this box in cubic centimeters. Solution (a) Calculate the volume as given: V = length x width x height V = 1.50 m x 1.50 m x 1.33 m = 2.9925 m3 (b) Write out the number of centimeters in a meter: 1 m = 100 cm (c) Write the conversion factor as a ratio:

cm 100

m 1

(d) Change the conversion factor to represent cubic meters (m3):

36

3

3

33

cm 10

m 1

cm 000 000 1

m 1

cm 100

m 1

(e) Multiply the volume by the conversion factor:

cm 10

m 1 m 2.9925 V

36

33

Notice that if one proceeds with this multiplication above, the desired units (cm3) will not be achieved since cancellation of units is not possible. (f) In the final step, the unwanted unit (m3) must cancel just as numbers would. Setting up this cancellation is the crucial step in unit conversion. To achieve the needed cancellation, the conversion factor must be “flipped”.

3

3

363 cm 500 992 2

m 1

cm 10 m 2.9925 V

Mathematicians have agreed on the following order to be used in performing a series of operations: 1. Simplify the expression within each set of

parentheses. 2. Perform exponents. 3. Perform the multiplications and divisions in order from left to right. 4. Do the additions and subtractions from left to right. “Please excuse my dear Aunt Sue” is a useful memory device for this order: parentheses, exponents, multiplication and division in order, and finally addition and subtraction in order. Measurements that have very large or very small values are usually expressed in scientific notation. Scientific notation consists a coefficient times the appropriate power of ten. The coefficient is a number equal to or greater than one and less. For numbers greater than one, the power of ten is positive. For numbers less than one, the exponent used is negative. To represent the number one, the exponent used is zero. The importance of scientific notation is that it allows for quick identification of the order-of-magnitude (power of ten) of a quantity. Calculations are often easier to perform when the values are listed this way and it removes any confusion in the number of significant figures.

1.4 Significant Figures There are two kinds of numbers used in science – counted or defined quantities and measured quantities. The value of a counted or defined number can be stated exactly. The number of chairs in a room or the number of coins in your pocket can be counted with certainty. Counted numbers are quantities that are defined as true and describe exact relationships. The number of centimeters in a meter or the number of seconds in an hour are examples. All measured quantities carry some uncertainty in their values no matter how carefully measured. The significance of a value has to do with whether it represents a true measurement or not. Any digit that is actually measured or estimated will be considered significant.


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Placeholders, or digits that have not been measured or estimated, are not considered significant. When working with the values of quantities it is important to keep proper account of the digits that are reliably known, such digits are called significant figures. Standard rules for writing and using significant figures: 1. In numbers that do not contain zeroes, all digits are significant. 2. Any zeroes between significant digits are significant. Examples: 0.705 kg (3 significant digits,

underlined) 2006 km (4 significant digits)

3. In a number that contains digits to the right of the decimal point, zeroes to the right of the last nonzero digit are significant. Examples: 37.0 cm (3 significant digits)

0.040 900 kg (5 significant digits) 4. When a number is expressed in scientific notation, all digits in the coefficient are significant. 5. In a number that has no decimal point and ends in one or more zeroes, the zeroes that end the number are NOT significant. Example: 40 s (1 significant digit) 6. Zeroes to the left of the first nonzero digit serve as placeholders for decimal point spacing and are NOT significant. Examples: 0.002 m (1 significant digit)

0.13 g (2 significant digits) The rules for working or calculating with significant figures depend upon the mathematical function. Multiplication and Division:

The number of significant figures in the result of a multiplication or division equals the number of significant figures in the factor containing the fewest significant figures.

Addition and Subtraction:

The significant figures in the result of an addition or subtraction are located only in places (hundreds, ones, tenths, etc.) that are reliably known for every value in the sum.

Important Notice: - To avoid excessive round off error, you

should only round to the proper number of significant figures at the very end of a calculation.

- If you need to use one of your answers in a

different part of a problem, use the original answer (not rounded).

1.5 Errors in Measurement A measurement is a means of expressing an observation with greater accuracy and precision. It provides a numerical value for an aspect of the object or event being observed. While measurement is the basis of scientific study, all measurements are approximate values based on limitations of measuring device, environmental conditions, measurement process and human error. It is important to understand that one error used in a mathematical formula leads to larger errors and misrepresentations. Errors are broadly classified in two categories:

systematic error

random error A systematic error results due to faulty measurement practices. The error is characterized by a deviation which is either less than or greater than the true value. Systematic error impacts the accuracy of measurement. When a measurement is close to the actual accepted value, the measurement is said to be accurate. To increase accuracy, use more sensitive equipment or repeat the readings taken. Systematic error results from: (1) instrument error - cannot be minimized or by repeated measurements since the same faulty tool will not improve accuracy. (2) procedural error - includes inappropriate physical environment, procedural mistakes and lack of understanding of the process of measurement. (3) personal bias - human habits which are not conducive for accurate measurement. A common bias leads to parallax or the distortion created when looking at an object at an angle.


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Random error unlike systematic error is does not occur in one direction. Some of the measured values are greater than true value; some are less than true value. It is possible to minimize this by repeating measurements. A random error is associated with precision which is a function of an instrument’s ability to measure values with greater detail due to smaller divisions. When the amount of error is expressed as a percentage, it is called percent deviation or percent error. It is a way for scientists to express inaccuracy expressing how far the experimental data is from the accepted measurement. When the absolute value of the answer is given all answers will be numbers without positive or negative signs. A negative percent deviation shows the experimental data is less than the accepted value while positive percent deviation shows greater experimental values. The formula below is generally used to calculate deviation:

100 x value accepted

value accepted from difference deviation %

1.6 Matter Classifications and Changes Matter is anything made of atoms and molecules and therefore takes up space and has mass. Everything is made up of matter. All matter is the same because all matter is made up of atoms. Matter is also different because objects can be made up of different kinds of atoms. Even though matter can be found all over the universe, it is only found in a few forms.

What makes a state of matter? The answer is the physical state of the molecules and atoms. Thus far there are five physical states of matter - solids, liquids, gases, plasmas, and Bose-Einstein condensates (BEC). Scientists have “always” known about solids, liquids, and gases. Plasma was noticed by William Crookes in 1879. The first BEC was produced by Eric Cornell and Carl Wieman in 1995 at the University of Colorado at Boulder. They later received a Nobel Prize for their work but the BEC was predicted in the 1920s by Satyendra Nath Bose and Albert Einstein. Each of these states is also known as a phase. Elements and compounds can move from one phase to another when specific physical conditions are present.

Generally, as the temperature rises, matter moves to a more active state or phase. A substance in a solid phase is relatively rigid, has a definite volume and shape and is often brittle. The atoms or molecules that comprise a solid are packed close together (a.k.a. dense) and are not easily compressible. Although the particles in a solid do not move freely about throughout the solid, they are in constant motion due to thermal energy; their atoms vibrate around fixed positions. The vibration is very small and rapid, and cannot be observed under ordinary conditions. There are crystalline solids whose atoms are arranged in regular repeating patterns and amorphous solids, such as glass, plastics, and wax that do not have a definite order or geometric pattern. Liquids are similar to solids in that their particles touch; however the particles are able to move around or flow at room temperature. Since the liquid molecules can move, they will take the shape of their container filling it up from the bottom first. While its shape may change due to flowing, its volume is definite. Another trait of a liquid is that of low compressibility because the distances between molecules are not large. The force of attraction among particles of a liquid is much greater than among particles of a gas, which is why a liquid does not need to be fully enclosed in a container. The particles in a liquid are in constant motion, their speed depending on temperature. Particles that have a greater than average speed can overcome forces of attraction, leave the surface of the liquid, and enter the gas phase. The term vapor is used to refer to the gas phase of a substance that is usually a solid or liquid at room temperature. Gases are transparent and have neither definite volume nor shape. Unconstrained gases will spread out indefinitely but if confined they will take the shape of their container and be evenly distributed. This is because gas particles have enough energy to overcome attractive forces resulting in a very low density due to well separated particles. With very little pressure, when compared to liquids and solids, those molecules can be easily compressed because of the empty spaces that exist between the

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molecules. The gas molecules move from an area of high pressure to one of low pressure. The term vapor and gas mean the same thing. Vapor is used to describe gases that are usually found as liquids at room temperature. Plasma is an ionized gas. It is a high temperature form of matter composed of ions and free electrons that can conduct electrical currents. Plasmas are the most common state of matter in the universe. They occur naturally in the universe in stars, quasars, and supernovas. On Earth, plasma is naturally occurring in flames, lightning and the auroras. Most space plasmas have a very low density making particle collisions unlikely; therefore they are usually referred to as collision-less. The collapse of the atoms into a single quantum state is known as a Bose-Einstein condensate is now considered the fifth state of matter. It can be thought of as the opposite of a plasma since it occurs at extremely low temperatures, close to “absolute zero” where the atoms are not moving at all. A Bose-Einstein condensate is all about molecules that are extremely close to each other (even closer than atoms in a solid). The density of a substance changes with changes in its phase. Almost all substances increase in density as they change from gas to liquid and from liquid to solid. Under ordinary Earth conditions, substances have their highest density as a solid because the atoms are closest together in that phase. Water is a common exception; it does not contract as its freezing point and its highest density is in the liquid state. Liquid water contracts until it reaches 4°C (actually 3.98°C but for this course 4oC is a good

approximation), then expands as it cools to 0°C. Therefore, water’s maximum density occurs in the liquid phase at 4°C. This occurs because of the unique properties of the water molecule and the structure of ice. This anomaly has important practical consequences. Since the densest water is at 4°C, when water at the surface of a pond or a river reaches this temperature, it sinks to the bottom. On the other hand, ice is less dense than water, so it floats on the water surface, forming a protective thermal layer. This prevents bodies of water from freezing all the way to the bottom, protecting aquatic wildlife and allowing them to survive the winter. Matter is constantly changing - it may change from one form into another by absorbing or releasing energy. In some situations it seems as if matter is disappearing, but the disappearance of matter is an illusion. For example, as a piece of paper burns it gives off heat and light energy. The matter is converted into carbon dioxide, water vapor, and other gases that escape into the atmosphere while some of the mass remains behind as ash. The same mass of each element is present before and after the change. Matter is neither created nor destroyed during these changes. A change that occurs in shape, form, or appearance but creates no new substance is called a physical change. A substance can move from one phase to another, but still be the same substance. For example, you can see water vapor over a boiling pot of water. That vapor can condense and become a drop of water. If you put that drop in the freezer, it would become a solid piece of ice. No matter what phase it is in, it is always water with the same chemical properties. Physical change affects one or more of the physical properties of a substance. This change does not alter the nature of the substance; it has the same molecular structure whether it is a gas, liquid, or solid. Since a physical change does not involve changing the chemical structure of the material, it can usually be changed back easily. Some examples are cutting, grinding, boiling, freezing, melting, condensing, breaking, separating, chopping, splitting, mixing, tearing, crushing and blending.

Figure 1-1: Compressibility of common phases of matters.

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A chemical change has occurred when a substance has changed into something new or different so that the original substance is gone. Chemical changes usually occur when two or more substances come in contact with each other, or when heat is applied. As matter and energy cycle through the environment, bonds between atoms and molecules are created or destroyed in different ways forming new substances. The new substances have different properties than the original material. Chemical changes represent chemical reactions and are usually written using chemical formulas such as:

2H2(g) + O2(g) = 2H2O This formula states that two hydrogen gas molecules react with one oxygen gas molecule to produce two molecules of water. Unlike physical changes, the products of a chemical reaction cannot easily be turned back into the original substances. Indications of a chemical change are change in color, production of gas (bubbles), production of a solid (precipitate), creation of heat (exothermic reaction), or cooling (endothermic reaction), light being given off and production of an odor. Not all chemical reactions show these indications in any easily visible way, and these changes do not necessarily indicate a chemical reaction. These are only visible evidences of activity that allow us to infer that a reaction is occurring on a molecular level. Some examples are digestion, combustion, spoiling milk or burning toast, iron rusting (iron and oxygen reacting), or acid fizzing when it comes in contact with limestone (acid and calcium carbonate reacting). A chemical change takes place in a battery to produce electricity.

1.7 Properties of Matter Substances are more easily identified when many properties have been recognized. Matter is sorted and classified according to its properties. All matter has properties. Properties describe matter. Some properties of matter never change and are used to identify unknown types of matter. These properties are called characteristic properties. All properties fall into

one of two categories: chemical and physical properties. Chemical properties have to do with the atomic or molecular composition of matter; they change the chemical nature of matter. Chemical properties are concerned with how substances react with other substances such as water, air or fire. These are properties that can only be observed by changing the identity of the substance. A piece of paper burns and turns to a black substance. After the flame goes out you can no longer burn the new substance. The chemical properties have been changed. Examples of chemical properties are heat of combustion, reactivity with water or other materials, and pH.

Physical properties, on the other hand, have to do with appearance. You can observe many physical properties with your senses and through measurement. Physical properties do not change the chemical nature of matter. There are many examples of physical properties and they are classified as either intensive or extensive. Intensive physical properties do not depend on the amount of the matter present in a sample while extensive physical properties do depend on the size of the sample. There are three basic quantities, length, time, and mass, which are essential and used to create other quantities such as surface area, speed, and density.

Intensive Extensive

Color Luster Solubility Malleability Ductility Density Viscosity

Odor Texture Hardness Magnetism Conductivity State of Matter

Volume Mass Length Weight

Temperature of Phase Changes

Length may be described as the distance between two points. The length of an object or the total length of a path an object moves is measured with a metric ruler or meter stick. Path length is usually measured in meters, but occasionally centimeters are more appropriate.

Table 1-4: Physical properties.


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You can convert a measurement in centimeters to meters by dividing by one hundred.

Most metric rulers and meter sticks used in class are calibrated - or scaled - in centimeters. The lines indicated by the numbers 1, 2, 3, and so on each represent a distance of one centimeter; the smaller divisions each equal 1 millimeter (10 mm = 1 cm).

Time is often described as our sense of things happening one after another or as the duration of an event. Elapsed time can be measured with a clock or stopwatch. One hour equals sixty minutes and one minute equals sixty seconds. Because many of measured events occur quickly, it may be necessary to record elapsed time to the nearest hundredth of a second.

The mass, or amount of matter contained in an object can be measured with an electronic balance or a triple-beam balance, which is a tool that works by comparing an object of unknown mass with an object of known mass. It is important that the balance be zeroed before determining the mass of an object. A mass that is determined in grams can be converted to kilograms by dividing by one thousand. A push or pull on a mass is called a force. Forces are measured with a spring scale. Ranges on spring scales typically vary from 2.5 newtons to 20.0 newtons. Mass should not be confused with weight, which is the pull of Earth’s gravitation on an object. The weight of an object may vary with its location, but its mass remains the same.

Temperature is often measured in degrees Celsius using a thermometer. The freezing point of water is 0°C; the boiling point is 100°C. A common unit for measuring angles is the degree (°), which is one ninetieth of a right angle. The protractor is an instrument used for measuring angles in degrees. A protractor that is a semicircle has a range from 0° to 180°, because a circle has 360°. If the sides of the angle to be measured are too short to intersect the edge of the protractor, they can be extended. Volume is the amount of space that an object occupies. It is a combination of three

dimensional quantities of length. The volume of solid objects can be determined by finding the volume of water an object displaces when it is placed in water in an instrument such as a graduated cylinder.

A graduated cylinder is often used to measure a liquid’s volume. Graduated cylinders are calibrated in milliliters (mL). Water and many other fluids form a meniscus (curving surface) when placed in the narrow tube of a graduated cylinder. To correctly read the volume of the liquid, place the cylinder on a flat surface and read from the bottom of the curved meniscus at eye level.

1.8 Vector Algebra Vectors form a very important part of the mathematical description of physics, so much so that it is absolutely essential to master the use of vectors.

Most of the quantities that we measure in everyday life are scalar quantities. A scalar quantity can be expressed by a number and an appropriate unit. It has magnitude (a numerical value) but no direction. Other physical quantities, called vector quantities, have a specific direction as well as magnitude. For example, we may need to specify that a car has traveled 50 km due south or that a plane is located 75 m southeast of the runway, or that a force of 20 newtons acts to the right rather than to the left.

Scalar Vector

Length Area Volume Mass Density Speed Temperature

Energy Entropy Work Power Pressure

Displacement Weight Velocity Acceleration Momentum Force

A two-dimensional vector can be represented graphically by an arrow in a coordinate system. The length of the arrow represents the

Table 1-5: Examples of scalar and vector quantities.


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Figure 1-2: The typical graphical representation of a vector in two dimensions.

magnitude of the vector quantity based on a specified scale. The direction of the arrowhead and its orientation with respect to the axes of the coordinate system provide the direction of the vector quantity.

In addition to a scale which will help determine the magnitude of the vector, the direction of a vector quantity is provided in terms of a reference point so that a reader will understand from which the angle should be measured. Reference points are very important in describing location. When determining the direction of a vector in a coordinate system, you must first understand the coordinate system used. If an x-y coordinate system is to be used, the answer will use terms such as “above the horizontal” or “from the positive y-axis”. For the purposes of this course, angle measurements should begin at the positive x-axis in an x-y coordinate system if not otherwise specified. If compass direction is provided, the answer for direction must include the appropriate compass direction. The “name” of the vector must describe its location in space. If a specific angle is not labeled in a given diagram, more than one name may be used to describe the location. For the purposes of this course, descriptions should include an angle that is measured with reference to the east direction unless otherwise specified.

In figure 4 two different vectors are shown (red arrow and blue arrow) and four different angles related to the vectors are labeled a – d. Angle a is measured to be 30o using the east direction as a reference point. It would be described as being N of E since one must move or turn toward the north when starting from the east direction to reach the red arrow. Notice that the red vector can also be described by using angle b as 60o E of N since to measure the 60o one must use a different reference point. Finally, the blue arrow can be described by using either angle c or d. Since both angles measure 45o, the description can be simplified as northwest. To correctly name any vector, one must know its magnitude, angle measurement, and angle description. If angle c is described as being 45o W of N, how would angle d be described?

1.9 Vector Mathematics Vector quantities are added and subtracted using geometric and algebraic methods. Vectors operate with scalar or vector quantities in a particular manner. Unlike scalar algebraic operations, vector operations incorporate a directional aspect. Any of the four basic functions (add, subtract, multiply, divide) can be performed using vectors, yet the method may not be as straightforward as with scalars. For example, multiplying a scalar by vector converts the scalar quantity into a vector without changing its magnitude, but assigning it a

40o Scale: = 1.0 m

Figure 1-3: Using the scale provided, a movement of 5.0 meters directed 40

o above the

horizontal axis is shown.

N

S

W E

d

45o

c 45

o

60

o

b a 30

o

Figure 1-4: Describing direction.


14

direction. Vector addition (and subtraction) is limited to vectors only. We cannot add a vector to a scalar.

The sum of two or more vectors is another vector called the resultant. The resultant vector is the result of the combined effect of the vectors. Like all vectors, the resultant has a magnitude and a direction. There are several methods for determining a resultant. The simplest case of vector addition occurs when the two vectors have a 0o angle between them. This means that the two quantities point in the same direction. The magnitude of the resultant is simply the sum of the magnitudes of the two vectors and the direction of the resultant is the same as the direction of the two vectors. The next case is also simple, adding vectors when the angle between them is 180o. This means the vectors are pointing in opposite directions. The magnitude of the resultant is equal to the difference of the magnitudes of the two vectors and the direction of the resultant is the direction of the larger vector. This leads to a special situation, the null vector. The null vector is conceptualized for completing the development of vector algebra. It is possible to encounter situations in which two vectors equal in magnitude but opposite in direction are added. The result of the algebraic operation has neither magnitude nor direction. In other words, the null vector is a vector whose components in rectangular coordinate system are all zero. The

null vector is denoted by the number 0 with the appropriate units of measure but no direction. These cases provide very important boundaries for answers. The magnitude of the resultant of any two vectors is a maximum when the vectors point in the same direction – there is a 0o angle between them. It is a minimum when the vectors point in opposite directions - a 180o angle between them. For any situation as the angle between two vectors increases from 0o to 180°, their resultant decreases from its maximum to its minimum value.

1.10 Adding Vectors Graphically There are two approaches to vector addition, a graphical method based on geometry and a component method based on algebra. The order in which two vectors are added does not matter. The act of moving vectors is perfectly correct if the vector’s length (magnitude) and orientation (direction) are not changed just its location in the coordinate system. The Parallelogram Method This method works well for a combination of two vectors only. In the parallelogram method, a diagram is drawn in which the vectors are positioned tail-to-tail. These original vectors will become adjacent sides of a parallelogram. (1) Create a scale for your diagram is the style of 1 cm = ___. (2) Draw the original vectors in a tail-to-tail fashion. (3) Construct a projection or a copy of one vector (vector A in figure 8) so that the copy begins at the head of the second vector (vector B) and is parallel to vector A. Name this projection vector A’. (3) Construct projection B’ so that it is parallel to

5.0 m 2.0 m

Resultant = 7.0 m

Scale: = 1.0 m

Figure 1-5: Addition of two vectors at an angle of 0

o.

5.0 m

2.0 m

Resultant = 3.0 m

Scale: = 1.0 m

Figure 1-6: “Addition” of two vectors at an angle of 180

o.

5.0 m

5.0 m

Resultant = 0.0 m

Scale:

Figure 1-7: The null vector – adding a vector and its opposite.


15

vector B and completed the parallelogram shape. (4) Draw the resultant R of the two vectors by creating a diagonal across the parallelogram – a line that points away from the place where the tails of the original vectors meet.

(5) Find the magnitude of the resultant by measuring the length of diagonal and using the scale. The direction can be found by measuring the angle between the “horizontal” and the resultant with a protractor and described with respect to the horizontal axis.

The Head-to-Tail Method This is a more useful method of addition since it works with combinations of two or more vectors. In the head-to-tail method (sometimes called the triangle law or method), a diagram is drawn to scale with the vectors positioned so that the tall of one vector is placed at the head of the other vector. The resultant of these two vectors, labeled R, is obtained by drawing a connection from the tail of the first vector to the head of the last vector. The magnitude of resultant is found by using the scale given and measuring the arrow’s length. The direction of R can be found with a protractor and described with respect to the horizontal axis.

1.11 Using the Pythagorean Theorem

If the vectors are at a 90° angle from one to another, the resultant can be found by using either of the graphic methods. It is more accurate and many find it is easier, to find the magnitude by using the Pythagorean theorem,

since the two vectors to be added form a right angle. The resultant is the hypotenuse of a right triangle. The Pythagorean theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides or a2 + b2 = c2. The direction of the resultant can be found mathematically by using trigonometric operations – in particular the inverse tangent function. The inverse tangent function finds the angle between the opposite and adjacent sides of the triangle:

side adjacant

side opposite

1-tan

b

a

1-tan

1.12 Vector Resolution The resolution of vectors is the reverse procedure of vector addition. A single vector is regarded as the resultant of two or more vectors, called its components, which must be determined. A vector can be resolved into an infinite number of components at a variety of angles. It is most useful to resolve a vector into

B

A

R

A’

B’

B

A

B

A

A’

B

A

A’

B’

Scale: 1 cm = 2 N

Figure 1-8: The parallelogram method.

Scale: 1 cm = 2.0 m

A

R C

B

Figure 1-9: The head-to-tail method.

a

c

b

90o

Figure 1-10: A right triangle.


16

two perpendicular components (one parallel to the horizontal axis and one parallel to the vertical axis in a coordinate system) so that simple vector addition may be employed. It is also important to realize that a component can be positive, negative, or zero. Since the components will be perpendicular to each other and originate from one vector, the result is a right triangle. Utilizing trigonometry the magnitude of the vertical and horizontal components can be found.

hypotenuse

side opposite sin

c

a

hypotenuse

sideadjacent cos

c

b

sideadjacant

side opposite tan

b

a

The magnitude of the horizontal or x-component can be found by using the formula:

Ax = A (cos ) where Ax the magnitude of the horizontal or x-component, A is the magnitude of the given

vector, and is the angle between the vector A and the horizontal. Similarly, the magnitude of the vertical or y-component can be found by using the formula:

Ay = A (sin ) where Ay the magnitude of the vertical or y-component, A is the magnitude of the given

vector, and is the angle between the vector A and the horizontal.

Question A person pulls a box across a floor by exerting a force of 60. N at an angle of 30.o to the horizontal. Find the horizontal and vertical components of the applied force. Solution (a) Sketch the situation. (b) Write out the formula needed to calculate the x-component.

Ax = A (cos ) (c) Solve the x-component.

Ax = (60. N) (cos 30.o) Ax = 52 N

(d) Write out the formula needed to calculate the y-component.

Ay = A (sin ) (e) Solve the y-component.

Ay = (60. N) (sin 30.o) Ay = 30. N

1.13 Using Vector Components If the components of a vector are known, the Pythagorean Theorem can be used to find the magnitude of the vector and the inverse tangent function can be used to find the direction. While this method can be used for one vector, it is particularly helpful when two or more vectors need to be combined. (1) A sketch the problem given. This does not require a scaled diagram. (2) Sketch the x- and y- components of each vector in the problem. (3) Using the component formulas, resolve each vector into its components. (4) Total all x-components into a single value usually known as xT. (5) Total all y-components into a single value usually known as xT. (6) Sketch a new diagram, not to scale, using the xT and yT values and showing the appropriate direction of each.

a

c

b

90o

Figure 1-11: Important trigonometric functions with a right triangle.

30.o

x

y 60. N


17

(7) Draw in the resultant of xT and yT. (8) Use the Pythagorean Theorem and the inverse tangent function to solve for the magnitude of the resultant and its angle measurement. (9) Use the revised sketch to describe the angle of the resultant and write the final answer in the accepted format.

Ax

Ay A

Bx

By B y

x

R

yT

xT

Figure 1-12: Examples of the suggested sketches when using the vector component to adding vectors.

Physics Honors Kinematics

18

Chapter 2 – Kinematics

Kinematics refers to the study of motion

of natural bodies. The bodies that we see and deal with in real life are three dimensional objects and essentially not a point object. A point object would occupy a point (without any dimension) in space. The real bodies or objects have dimensions, length, width and height. A real body cannot be specified by a single set of coordinates. A second equally important aspect is that different parts of the bodies may have paths or trails different to each other. When a body moves with rotation (rolling while moving), the path of different parts of the bodies are different; on the other hand, when the body moves without rotation (slipping/ sliding), the paths of the different parts of the bodies are parallel to each other. In the second case, the motion of all points within the body has equivalent translational motion. Therefore, it is possible to treat the body as a point so long rotation is not involved. For this reason, kinematics is divided into two studies:

translational kinematics

rotational kinematics A motion can be a combination of translational and rotational movements or solely or type or the other. Translational motion allows an object and all of its particles as one point, whose position can be represented by a single set of coordinates. This chapter will concentrate on translational motion. 2.1 Position, Distance, and Displacement Motion has no meaning without a reference system. Any description of motion takes place in a coordinate system that allows us to track the position of an object. One-dimensional or straight-line motion means that objects are only free to move back and forth along a single line. As a coordinate system or frame of reference, we usually choose an x-axis with a specified origin and positive and negative directions. The location of the origin and which way is called positive or negative may be chosen according to the problem. Conventionally, positive values are

assigned to objects moving up or the right as compared to the origin or start of the problem and negative values to objects moving downward or to the left. In daily life, motion is recognized with respect to an observer or a stationary object. If the object maintains its position with respect to the stationary objects, we say that the object is at rest; else the object is moving with respect to the stationary objects. It is important to realize that all objects on Earth are actually always in motion since the planet itself is continually rotating and revolving. It is accepted that all objects moving with the Earth without changing their positions on its surface are stationary objects in the Earth's frame of reference. When traveling on aircraft, one is hardly aware of the speed of the aircraft. Fellow passengers and parts of the airplane are all moving at the same speed, giving the impression that passengers are simply sitting in a stationary cabin. One is alerted to actual movement when looking out the window and seeing passing clouds and a changing landscape. Consider two persons standing on a subway platform. They both perceive the motion of a passing train in exactly the same manner since they are in the same state of motion. On the other hand, a passenger in a speeding train finds that the other train crossing it on a parallel track in opposite direction has the combined speed of the two trains (v1 + v2). The observer on the platform finds them running at their individual speeds v1 and v2. It is important to realize that motion is an attribute which cannot be stated in absolute terms; it results from the interaction of the motions of the both object and observer (frame of reference). The distance traveled by an object that moves from one position to another is a magnitude of motion in terms of the total length of travel during the trip. This total length of travel depends on the path taken as the object moves from its initial position to its final position. It does not matter if the object goes further away or suddenly moves in a different direction or


19

reverses its path. The magnitude of movement keeps adding up so long the object moves. Distance is not linked with any directional attribute therefore it is a scalar quantity of motion which is cumulative in nature and does not take on a negative value. Since distance is the measurement of length, its SI measurement unit is the meter and its symbol is usually d and sometimes s. Initial and final positions of the object are merely start and end points of measurement and are not enough to determine distance. It must be understood that the distance is measured by the length covered, which may not necessarily be along the straight line joining initial and final positions. This fact distinguishes distance from displacement, which is the change in position of the object regardless of the path taken. In other words, a vector quantity that is equal to the length and direction of the shortest straight line joining an object’s initial and final positions. The primary difference between distance and displacement is that the distance an object travels tells you nothing about the direction of travel, while displacement tells you precisely how far, and in what direction, from its initial position an object is located.

2.2 Speed and Velocity An important part of describing motion is to specify how rapidly an object moves. Average speed is a scalar quantity that is the distance traveled divided by the amount of time it took to travel that distance. Another, sometimes more appropriate, way to describe the rate of motion is to find an average velocity. In English, speed

and velocity are sometimes used interchangeably but in science, the difference between speed and velocity is very important. Speed relates to the distance traveled and is a scalar quantity while velocity relates to displacement and is therefore a vector. Average velocity only relates to the rate at which an object goes from its initial to final positions regardless of the path taken, and thereby specifies direction so it is a vector quantity. The SI unit of measurement for both speed and velocity is meters per second (m/s). The direction associated with a velocity would be the same as its displacement. The most commonly used formula for average speed or velocity is:

t

d v

where v , usually written as v, represents the average speed or velocity of the object, d the distance the object traveled, and t the time needed to travel that specific distance. To find the simple average speed or velocity of an object, use the following formula:

2

v v v f i

where vi is the initial speed or velocity of the object and vf is its final speed or velocity. Some situations require more than just the average rate of motion. The instantaneous speed refers to the speed of an object at a specific instant in time. Instantaneous velocity is the speed and direction of an object at a particular instant in time. The case of uniform velocity is special in that the average velocity over any time interval equals the instantaneous velocity at any time. The speed and direction of the moving object are constant.

2.3 Acceleration Objects can move at changing speeds and objects can also change direction while moving. Acceleration is the rate at which an object's velocity is changing. This velocity could be

Figure 2-1: Distance vs. displacement.


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Figure 2-2: Types of motion. (a – b) Constant motion. (a) Rest – no movement. (b) Uniform speed – moving with one non-zero speed. (c – d) Accelerated motion. (c) Positive acceleration – speeding up. (d) Negative acceleration – slowing down.

changing because the object is slowing down, speeding up, turning around, or any combination thereof. If there is any change in the speed and/or direction of motion of an object, it is accelerating. An important special case of accelerated motion occurs when the acceleration is constant or uniform. This simply means that the rate at which the velocity changes (increases or decreases) is the same at every instant in time during the motion. It is important to realize that the sign (positive or negative) of the acceleration does not denote the direction of an object’s movement. In general when an object experiences horizontal linear motion, the object in question will slow down if the velocity and acceleration have opposite signs (one positive and the other negative). When the velocity and acceleration have the same sign, the object will speed up. If an object’s speed increases, the change in speed is referred to as positive acceleration. If the object’s speed decreases, acceleration is negative and the object can be said to decelerate.

The unit for acceleration is m/s/s (meters per second per second). This indicates that acceleration is a change in velocity or speed (in m/s) per unit time (in s). Using standard rules of algebra, the units can be combined to form m/s2 (meters per second squared).

A simple formula for acceleration is:

vf = vi + at where a represents the object’s acceleration. Sometimes this formula is stated as:

t

v a

2.4 Vertical Motion All objects near Earth are pulled toward Earth by gravity. Earth’s gravity causes falling objects to accelerate, that is, the speed of the object increases as it falls toward Earth’s surface. When air resistance is negligible or ignored, objects near Earth's surface fall with a constant acceleration equal to the acceleration due to gravity. The symbol g represents the magnitude of this acceleration, which has an average value of 9.81 m/s2. The downward direction of gravitational acceleration can be taken to be positive since it increases the object’s speed. The acceleration of the object is the same throughout its motion in both its magnitude and direction. Objects undergoing this type of motion, when gravity is the only important influence, are said to be in free fall. Ignoring the effect of air resistance, it does not matter how massive the object is or what it is made of. An object allowed to drop has a velocity that increases until it hits the floor. In reality, an object that has a large surface area relative to its mass is affected more by air resistance than an object with a smaller surface area relative to its mass. The equations of motion apply as if the objects move in a vacuum.


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An object launched directly upward, without any motion left or right, from a given height takes an equal amount of time to reach the top of its path as it takes to fall from the top of its path back to the height from which it was launched. The velocity an object has at a given height on its way up is equal in magnitude but opposite in direction to the velocity it will have at that same height on its way back down. Usually the upward velocity is considered to be positive and the downward velocity negative. When the object reaches its highest point, called the apex, its instantaneous velocity is zero. Two important formulas that can be used to find the time needed to reach the apex or the height of the apex are:

d = vit + ½at2

vf2 = vi

2 + 2ad These formulas can serve a variety of purposes. The first can be used to find the location (distance traveled) of the object at any time during its flight. The second can be used to find the maximum height traveled by an object when making vf equal 0 m/s. When using these formulas it is important to remember that the acceleration of a “free-falling” object is the acceleration due to gravity (9.81 m/s2 on Earth). For an object that is thrown vertically upward, the object “fights” gravity on its

way up so the value used for acceleration is negative because it slows the object.

2.5 Graphical Analysis of Linear Motion

The motion of an object is often analyzed graphically. Graphical analysis is useful for many things and can be used to determine what kind of motion is being observed. In order to do that we must first know what kinds of graphs the different types of motion produces and how to obtain information from them. A position-time graph can be used to determine an object’s velocity and position by plotting time on the horizontal axis and position of the vertical. Many times a position-time graph is referred to as a distance-time graph. Distance-time graphs simply plot the two scalar quantities along two axes. However, there are certain restrictions:

distance is a positive scalar quantity therefore graphs occur in the first quadrant.

as distance increases during a motion, the slope is always positive.

when motion stops, the plot becomes a straight line parallel to the time axis so that distance is a constant value.

Plotting displacement on two axes is challenging. For the purposes of this course it is simplified to one dimensional motion having only two possible directions, along or opposite to the positive direction of axis. Therefore once the “forward” motion of the object is determined in the problem,

Figure 2-3: Free fall in air vs. in a vacuum.

Figure 2-4: Distance-time graph.

Dis

tance (

m)

Time (s)

Distance is a constant value; object has stopped moving.

Distance values are changing; object is moving with constant speed.


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the object can be shown to reverse its direction of motion by moving in the opposite direction or “crossing” the +x-axis and continuing into the fourth quadrant. A distance-time or displacement-time graph can be used to find an object’s speed or velocity, respectively, by calculating the slope between two time intervals. The slope is defined as the rise divided by the run of a section of the graph. Any two points on a graph can be connected by a straight line. It is important to realize that constant slopes indicate constant speeds (or velocities) while changing slopes indicate changing speeds (or velocities) also known as accelerated motion. A line with a changing slope is usually curved. For any given point on the curve there is a line, called the tangent line that intersects the curve at one point. Calculating slope of the tangent would result in an instantaneous value for the property needed.

A velocity-time or speed-time graph is obtained by plotting data for rate on the vertical axis and time on the horizontal axis. The difference between the two graphs is that a speed-time graph is located in the first quadrant whereas a velocity-time graph may cross into the fourth quadrant if the direction changes during the time interval plotted. An acceleration-time graph has acceleration on the vertical axis and time on the horizontal axis. Most acceleration-time graphs in this course will represent instantaneous accelerations and will therefore have an abrupt look formed by straight horizontal and vertical lines. The three graphs of related through their slopes. An analysis of the three types of graphs shows that the slope of a position-time graph is the

Figure 2-5: Displacement-time graph.

Object is traveling forward at a constant speed.

Object is traveling “backwards” with a constant speed.

Dis

pla

cem

ent (m

)

Time (s) 0 Dis

tance (

m)

Time (s)

Increasing speed

Decreasing speed

Constant speed

Figure 2-7: Represents non-uniform velocity; the slope is not a straight line.

Figure 2-8: Straight line slope equals uniform acceleration.

2m/s 1.5a

s 4.0

m/s 6.0

t

v

a

Figure 2-6: Slope equals velocity on a displacement- time graph – in this case, a constant velocity.


23

object’s speed. The slope of a velocity-time graph is the object’s acceleration. The area under the velocity-time graph, between the vertical axis at each time point, is equal to the distance traveled by the object in that time. This information can be used to construct a distance-time graph. The area under an acceleration-time graph can be used to construct a velocity-time graph. For the cases of constant velocity and constant acceleration, these areas are rectangular and triangular. For the purposes of this course, calculation will be limited to those based on the areas of these simple shapes rather than the use of calculus. The area formulas are listed below:

A = ½bh

where b represents the base of the triangular shape usually the horizontal axis value and h represents its height, usually the vertical axis value.

A = lw where l represents the length of the rectangular shape usually the horizontal axis value and w represents its width, usually the vertical axis value.

2.6 Two-Dimensional Motion An object that is thrown, shot, or otherwise projected into the air is called a projectile. Projectiles move in a very different way than free falling objects. When you throw a baseball, the ball first climbs and then falls accelerating downward due to gravity. It does not fall straight down; rather it falls in a curved path. The flight path of a projectile is referred to as its trajectory and takes the shape of a parabola. All of the time the projectile is moving forward at a constant speed (ignoring air resistance), gravity is accelerating it down toward Earth’s surface. This is called two-dimensional motion. When an object is propelled in the horizontal direction only without any initial movement up or down, it is said to be launched horizontally. The object leaves your hand or a machine with a constant horizontal velocity and since air resistance is ignored, the horizontal velocity is constant at all times during the flight. As the object moves horizontally, gravity acts to pull it down. Gravity does not and cannot affect the horizontal movement. The object proceeds to

Figure 2-9: Analyzing motion graphs. If one was provided the top graph, the slope can be calculated and found to equal 4 m/s. Using the constant speed of 4 m/s, one may plot a v-t graph (center). Finally, based on the slope of the center graph (0 m/s

2), the acceleration-time graph

(bottom) can be plotted.

Figure 2-10: Area under a velocity-time graph equals the distance traveled – in this case 80 m.


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move both ways – horizontally and vertically – resulting in the characteristic curved path. Knowing that two-dimensional motion is really two separate cases of one-dimensional motion makes it easier to find and use mathematical equations. It is important to know that the time of flight is controlled by the vertical height above the ground, not by the horizontal speed. Once the time of flight is known, you can find the range of the object (horizontal distance traveled) as well as the speed and direction at any point in the path. Since the object is falling in a curved path, the object’s final velocity must be found using the Pythagorean Theorem since it will have both an x and a y component. The range of an object is found using:

dx = vxt where dx is the horizontal distance traveled by the object or its range, vx is the horizontal speed with which it was thrown, and t is the time the object was in the air. Using the time needed to fall to the ground and the acceleration due to gravity, one may find the height from which the object was thrown:

dy = viyt + ½at2 where dy is the height, viy is 0 m/s since the object was horizontally thrown (not given any vertical initial speed), a is the acceleration due to gravity, and t is the time needed to fall. When any object is launched horizontally on Earth, it will begin to fall under the influence of gravity and pick up vertical speed. To find the vertical speed at any time during the object’s descent, use the following formula:

vy = viy + at

where vy is the vertical speed of the object, viy is again 0 m/s, a is the acceleration due to gravity (9.81 m/s2 on Earth) and t is the specific time of interest. Once the vertical speed at a specific time is found, one may use the Pythagorean Theorem and inverse tangent formula to find the total velocity (magnitude and direction) of the falling object at any given point.

vf2 = vx

2 + vy2

xv

yv 1-tan

where vf is the object’s speed at a specific location, v is the horizontal speed the object was given when thrown, and vy is the vertical speed of the object at a specific point in time.

2.7 Projectiles at an Angle When an object is launched upward at an angle it is given an initial velocity in both the horizontal and vertical directions. Only the horizontal component of velocity applies to the horizontal motion and only the vertical component of velocity applies to the vertical motion. The individual x and y components of the velocity can be found using the formulas below:

vx = vi cos

vy = vi sin where vx and vy are the x and y components of

the initial velocity, vi is the initial velocity and is the angle formed between the initial velocity vector and the horizontal. Just as with horizontal launches, the horizontal velocity is constant at all times during the flight when air resistance is ignored. The vertical velocity is slowed by gravity until it stops at the highest point in the object’s trajectory.

Figure 2-11: A horizontally launched projectile.


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Figure 2-12: Example of an angular launch.

Figure 2-13: Two points in a projectile’s trajectory: point A shows the object rising

toward the apex while point B shows the object descending.

While there are many situations that may occur in real-life, most problems in this course will deal with objects that have the same initial and final elevations. The launch angle that produces maximum range under these conditions is 45o. If the launch and landing points are at the same height or level, the time it takes a projectile to reach its apex equals the time it takes to fall from that height back down. Since the effects of air resistance are ignored in this course, the time needed to reach the apex is doubled to find the total time of flight. It is also important to remember that the speed of a projectile at a given height on its way up is equal to the speed it will have at that same height on its way back down. It is important to realize that the projectile’s velocity can be calculated at any given point by employing the Pythagorean Theorem and inverse tangent function. While the projectile is traveling upward, the velocity is a positive value and the acceleration due to gravity is negative since it is slowing the object. Once the projectile passes the “halfway” point in its trajectory and is heading toward the ground, the velocity is negative to denote downward and the acceleration due to gravity is positive.

Physics Honors Statics and Dynamics

26

Chapter 3 – Statics and Dynamics

3.1 Balanced vs. Unbalanced Forces A force is a push or a pull. A force can stop motion, change the speed of motion, and change the direction of motion. When the forces acting in an object are balanced, net force equals zero, the object is at rest. If forces become unbalanced, by the introduction of other forces, an object moves. Precisely how an object moves in response to a force depends on the object's mass. Mass can be thought of as a measure of the matter content of an object. More importantly for motion, mass is a measure of an object's natural tendency to move with constant velocity, referred to as its inertia. That is, mass is a measure of inertia. An unbalanced force always produces motion. This means that the resultant force was strong enough to move an object in some direction. If the combinations of forces acting on the object had been balanced somehow, the cart would not have moved. When a set of balanced forces acts on an object but does not move it, the object is said to be in a state of equilibrium. Where two forces act in the same direction, it is easy to see that a force acting in the opposite direction to that of the resultant force would produce a state of equilibrium. A force that balances the resultant force is called an equilibrant force. Where the two forces act at an angle to each other the equilibrant force must be equal in magnitude to the resultant force and must act in a direction exactly opposite to that of the resultant force.

3.2 Newton's First Law of Motion Newton's first law of motion is also known as the law of inertia. This law states that an object at rest will remain at rest and an object in motion will remain in motion until an outside force acts on the object. There are two parts to this law. First, any object at rest will not move unless some force acts on it. Second, any moving object will continue to move unless a force acts on the object to slow it or to change its direction. This law says that the natural state of motion is that of constant velocity – it does not distinguish between a constant velocity of zero (rest) from any other constant velocity. Inertia is the tendency of an object to resist any change in motion. The more massive an object is, the more inertia it has, or the more it resists a change in motion. A misconception is that to keep an object moving, a constant force must be applied. This is false since an object does not come to rest when a force is removed but due to friction between surfaces.

Figure 3-1: Balanced forces.

Figure 3-3: Law of Inertia.

Figure 3-2: Equilibrant force.


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3.3 Newton's Second Law of Motion

Newton's second law of motion is also known as the law of acceleration. This law states that the acceleration of an object depends on the ratio of the acting force to the object’s mass. An increase in force on a given mass will increase acceleration. If an equal force is applied to two objects of different mass, the object with the lesser mass will have the greater acceleration. An object with a large mass will not be moved easily by a small force, whereas a large force acting on the same object will move it more easily. Therefore, mass is what determines how strongly an object "wants" to keep its velocity constant. A misconception is that larger amounts of force are needed to continue the motion of a moving object. This is false since once an object reaches a constant speed, you need only to balance friction to continue moving. You no longer need to accelerate the mass of the system. The most common formula for Newton’s second law of motion is:

F = ma where F is the force, m is the mass of the object and a is the acceleration of the object caused by the force applied to it.

3.4 Newton's Third Law of Motion Newton's third law of motion is also known as the law of action and reaction. This law states that for every force applied to an object, there is a force equal in magnitude and oppositely

directed applied by the object back onto the original agent. An important point to remember here is that a force and its reaction always act on different object. Therefore, these forces never cancel each other. Basically, this law says that a single object cannot act upon others without being acted upon; two objects always interact applying equal and opposite forces to each other.

3.5 Types of Forces In situations involving force, the use of a free-body diagram can be important. In such a diagram, the object is isolated making it a “free body," then all the force vectors acting on the object are drawn. In most cases the object is represented as a point. A good general strategy for drawing a free-body diagram is to remember that forces occur in pairs. Many different kinds of forces exist. However, they can be grouped into two broad categories: forces that result from contact between two objects and forces that can act at a distance between objects that are not in contact. These forces are known as contact forces and field forces, respectively.

Figure 3-4: Law of Acceleration.

Figure 3-5: Law of action and reaction.

Figure 3-6: Example of a free-body diagram moving left.

Fa Ff

Fg


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Field Forces Weight or the force of gravity (Fg or W) is a direct result of Earth's gravitational pull; in fact, on Earth, a body's weight is the downward gravitational force exerted on the body by Earth (toward the center of Earth). The greater the mass, the greater the force of gravity. The closer the masses, the greater the force of gravity. The fact that a mass exerts a force on another mass some distance away means that the space between the masses is unlike that where no masses are present. We say that every mass sets up a gravitational field around itself. The field acts on other masses so that an attraction results. A gravitational field is a vector quantity with the direction of the field being the direction of the force exerted on the mass. The formula used to find the weight of an object is:

Fg = mg where Fg is the object’s weight, m is its mass in kilograms, and g is the acceleration due to gravity (9.81 m/s2 on Earth). Electrical force is the force of attraction between unlike charges or the force of repulsion between like charges. Magnetic force is a force of attraction or repulsion that comes into existence when electric charges are in motion. Unlike magnetic poles attract, and like poles repel. Electrical and magnetic forces are will be discussed in detail later in the course. Contact Forces The applied force (Fa or Fapp) is the force which is exerted on an object by a person or another object in a specific direction. Tension (FT or T) is a force transmitted through a rope, string, or wire when it is pulled tight by forces acting on either end. The tensional or tensile force acts along the rope, etc. and pulls equally on objects at either end of the rope. A spring force (Fs) is one exerted by a compressed or stretched spring upon any object that is attached to it. An object that compresses or stretches a spring is always acted upon by a force that restores the object to its rest or

equilibrium position. Spring forces and elasticity will be addressed later in the course. The normal force (Fn) is a supporting force exerted by a stable object such as a surface in response to another object that is in contact with it. It is always perpendicular (normal means perpendicular) to the surface between the objects. The normal force that the floor exerts on your feet when you are standing on a horizontal surface like a floor is equal to your weight. The normal force between surfaces is an important factor in determining the friction between those surfaces.

When two surfaces are in direct contact and one surface either moves or attempts to move across the other, a force of friction (Ff) that opposes the motion is generated between the surfaces. The origin of friction is based on the microscopic irregularities of the surfaces involved and intermolecular forces of attraction. Friction depends on the nature of the surfaces and how hard they are pressed together. Making surfaces smoother or using lubricants can reduce friction. Lubricants separate the contact surfaces. The use of ball bearings is often effective because rolling friction is generally weaker than sliding friction. Friction can be found using the following formula:

Ff = µFn

where Ff is the force of friction, µ is the coefficient of friction (a property of the surfaces involved), and Fn is the normal force. There are two main types of friction: static (Fs) and kinetic (Fk). Since the subscript become cumbersome and confused with other symbols, they are usually not used in this course therefore any friction is referred to using the generic Ff. Static friction is the frictional force that opposes any attempt to move a stationary object along a surface. It is represented as an inequality of the formula shown above since it varies to a maximum value which depends on the surfaces in contact. It is not dependent upon surface area.


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Kinetic friction, which is more accurately called sliding friction, is the frictional force that opposes the sliding motion of two surfaces rubbed together. As with static friction it is independent of surface area and it is also independent of speed. One important fact to remember is that the “kinetic” friction between objects is always less than static friction between the same objects. Air resistance or drag is a specific example of the force of friction between moving objects and the air. Usually it is small enough to be ignored but becomes significant at high speeds and for objects with large surface areas.

3.6 Inclined Surfaces Life isn't conveniently arranged to take place only on smooth, flat, horizontal surfaces. Sometimes you just have to go uphill or downhill. In these situations, it is best to choose a coordinate system with axes that are parallel and perpendicular to the surface. Generally, the x-axis is parallel to the surface of the incline and the y-axis is perpendicular to the surface. It is important to remember that the weight of an object always acts vertically downward, can be resolved into x- and y-components. The normal force is always perpendicular to the surface the object rests upon so it is represented by a component of the object’s weight. Due to geometry, the angle the weight makes with the inclined surface is the same as the angle of the incline.

3.7 Momentum The momentum of an object is defined as the product of its mass and its velocity:

p = mv where p is momentum. Momentum is a vector quantity whose direction is the same as that of the velocity. In a way you can think of momentum as a measure of the effect the motion of an object has when that object interacts with other objects. A fast moving car has more momentum than a slow moving car of the same mass. A heavy bus has more momentum than a small car moving at the same velocity.

The SI unit of momentum is a kilogrammeter per second which has no special name but is

sometimes abbreviated as a newtonsecond. Newton's second law was studied earlier in this course and was applied to circumstances in which the mass remained constant. However in the many cases the mass of a system may change during the motion. The most general form of Newton's second law expressed, in terms of momentum:

ma t

mv -mv F if

3.8 Impulse The average force applied to an object times the amount of time this force is applied is called the

Figure 3-7: Example of an inclined plane.

Fn

Fg

Fg cos

Fg sin

Figure 3-8: Forces on an inclined plane.


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impulse. Impulse is a vector quantity whose direction is the same as the net force acting on the object. When a net force acts on an object, the object’s velocity and momentum change.

The SI unit of impulse is the newtonsecond. The same impulse can be delivered by a weak force acting for a long period of time or a strong force acting for a short period of time. The concept of impulse is closely related to momentum by what is often called the impulse-momentum theorem. An impulse applied to an object causes a change in the object's momentum. If there is no impulse then there is no change in momentum:

J = Fnett = mv where J is impulse.

3.9 Conservation of Momentum The law of conservation of momentum states that if no external force is acting on a system, the total momentum of the system remains unchanged. This means that when a set of objects interact, the total momentum before the event equals the total momentum after the event. The net impulse on an object will be zero when the net force on that object is zero. When the net force on an object is zero, its momentum is conserved. Since momentum is a vector quantity it must be conserved in both magnitude and direction.

3.10 Collisions The concept of momentum is important when two objects interact. We will focus on an interaction called a collision. A collision occurs when the forces of interaction between two objects are large for a limited period of time. Although the forces in an action-reaction pair may be equal and opposite, the masses of the two objects to which the forces are applied may not be equal. In a collision between a massive bowling ball and a much less massive bowling pin, the ball exerts a force onto the pin and the pin exerts an equal and opposite force onto the ball. The action force of the bowling ball produces a big change in velocity of the bowling

pin. The reaction force of the pin barely changes the velocity of the ball. Collisions are often divided into two categories according to whether or not the total kinetic energy of the colliding bodies is conserved. Inelastic Collisions If the total kinetic energy is not conserved, the collision is called an inelastic collision. To analyze these types of collisions we just apply the conservation of the momentum to the system. A special case of an inelastic collision occurs when the colliding objects stick together and emerge from the collision effectively as one object - this case is called a completely inelastic collision because the system loses the maximum amount of kinetic energy that is can lose while still conserving momentum. The formula for the inelastic collision of two objects is:

(mv1)i + (mv2)i = (m1 + m2)vf

Elastic Collisions A second category of collisions considers situations when total kinetic energy is conserved. These collisions are called elastic collisions. As with momentum, it is the total kinetic energy of the system, not of any individual particle, that is conserved. In everyday life few collisions are precisely elastic. However, many everyday collisions are almost elastic because such a small fraction of the kinetic energy is lost. In a head-on collision, we can use the equations for the conservation of momentum and the conservation of kinetic energy to solve for the final velocities of the two particles. The formula for the elastic collision of two objects is:

(mv1)i + (mv2)i = (mv1)f + (mv2)f


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3.11 Uniform Circular Motion Imagine a rocket launched horizontally. If there were no gravity, inertia would cause the missile to fly horizontally until some external force stopped it. Since there is gravity the rocket is pulled toward Earth as the rocket moves horizontally, the path is curved downward and the rocket eventually strikes the surface. If a powerful enough engine is used in the rocket, the rocket would travel far enough horizontally to not strike the Earth when its path is curved downward; the rocket would be in orbit around Earth. An interesting and important case of two-dimensional motion is provided by an object that circles around a point at constant speed. The object is said to undergo uniform circular motion. The circling object is always traveling in the direction of the tangent to the path at the object’s location. Although the magnitude of the object’s velocity is constant, the velocity direction is constantly changing. A change in the velocity vector means that the object is accelerating. The acceleration experienced by an object in uniform circular motion is called centripetal acceleration. Centripetal acceleration is a vector quantity directed toward the center of the circle and must be perpendicular to the velocity; the formula is:

r

v2

c

a

where ac is centripetal acceleration, v is the speed of the object in the circular path, and r is the radius of the circular path. According to Newton's second law, where there is an acceleration there must be a force that causes it. Since a circling object is accelerating, there must be a net force acting on it – it is not in equilibrium. The force that causes the centripetal acceleration is called the centripetal force. This force acts in the same direction as the centripetal acceleration – toward the center of the circle. It is important to recognize that centripetal force is not a new kind of force; a formula:

r

mvF

2

c

where Fc is centripetal force.

3.12 Law of Universal Gravitation

The ancient natural philosophers believed that objects fall because Earth pulls on them. Isaac Newton proposed that Earth cannot be the only object to attract objects. All matter must attract other matter - he claimed the attraction is universal. Gravitational force or gravity is a force of attraction between objects that depends on the mass of the objects and the distance between them. The gravitational forces between ordinary objects are extremely weak and therefore often undetectable. However, if one or both of the objects contain an enormous amount of mass, the gravitational force between them is significant. The gravitational force between Earth and an object on its surface is also strong

Figure 3-9: Rocket in orbit.

Figure 3-10: Uniform circular motion.


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enough to be noticed. It is referred to as the object’s weight. Newton's law of universal gravitation describes the behavior of gravity. This law applies only to (1) masses whose sizes are small compared to the distance between them (point masses); and (2) spherical masses of uniform density – when the distance between them is measured from the center of one to the center of the other. The formula is:

2

21

r

mGm F

where G is the universal gravitational constant

(6.67 x 10-11 Nm2/kg2), m1 and m2 are the two masses, and r is the distance between the masses. Objects near Earth’s surface are a distance from the center roughly equal to the radius of the planet. Using Newton's law of universal gravitation we can calculate the acceleration due to gravity near any satellite’s surface using the formula:

2r

Gm a

where m and r both refer to the planet or large body. The outcome is that the higher an object is above the surface, the further from Earth's center it is, and therefore, the weaker the effect of gravity resulting in a smaller acceleration.

3.13 Kepler's Laws of Planetary Motion One of the great early successes of Newton's work on gravity was his ability to use his result to accurately explain the laws of orbital motion that Johannes Kepler discovered from his observations of Mars and the other known planets. These laws of orbital motion are summarized as three statements: 1. Law of Orbits - Planets follow elliptical orbits, with the Sun at one focus of the ellipse.

The Sun is not at the center of the ellipse, but is instead at one focus (generally there is nothing at the other focus of the ellipse). The planet then follows the ellipse in its orbit, which means that the distance between the planet and the Sun is constantly changing as the planet goes around its orbit. The point of nearest approach of the planet to the Sun is termed perihelion; the point of greatest separation is termed aphelion. 2. Law of Areas - As a planet moves in its orbit, a line from the Sun to the planet sweeps out equal amounts of area in equal amounts of time. The line joining the Sun and planet sweeps out equal areas in equal times. A planet completes its motion with constantly changing velocity as it moves about its orbit. The planet moves fastest when it is near perihelion and slowest when it is near aphelion. 3. Law of Periods – A planet’s period increases as its distance from the Sun increases in a specific ratio as a consequence of the gravitational force. The term period refers to the time needed to complete one revolution.

Figure 3-11: Kepler’s first law of motion.

Figure 3-12: Kepler’s second law of motion.


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3.14 Moment of Force Parallel forces act together along parallel lines at different points on the same object. They may act in the same or in opposite directions. Whenever parallel forces acting in one direction are balanced by one or more parallel forces acting in the opposite direction, the resultant force becomes zero. Parallel forces may act in the same direction on an object that is balanced on a pivot, or fulcrum, and is free to turn, or rotate, about the fulcrum. If the parallel forces are equal, the balanced object remains balanced and does not move. If the forces are unequal, the object is unbalanced and begins rotating in the direction of the larger of the two forces. The tendency of a force to produce rotation of an object is called the moment of force, or torque. If the rotation is toward the right, the rotation is said to be clockwise; if in the opposite direction, it is said to be counterclockwise. The rotation caused by unbalanced moments of force is seen when two children of unequal weight are placed on a see saw at equal distances from the fulcrum. The moment of a force depends not only upon the magnitude of the force but also on the distance of the force from the fulcrum. The formula used to calculate torque is:

) sin (F d

where is the torque, F is the force applied, d is the distance between the force and the fulcrum,

and is measured between the force and the distance. When the clockwise and counterclockwise moments of force are equal, an object balanced

on a fulcrum and does not rotate - the object is in equilibrium. The relationship expressed in the law of moments enables you to find the weight of an unknown object when its distance from the fulcrum is known if you have another known weight whose distance from the fulcrum is also known. Commonly the law is expressed as:

right

left

where denotes the term sum.

Figure 3-13: Force exerted on one side of a see-saw causes it to move.

d

Physics Honors Work and Thermodynamics

34

Chapter 4 – Work and Thermodynamics

Matter and energy are what make up the universe. Matter is the substance of the universe. The concept of matter is fairly simple to understand. Matter has mass and takes up space. You can see, hear, feel, taste, and smell matter. The concept of energy is one of the most important ideas in science but is more difficult to understand. Energy is considered to be the “mover of matter.” In other words, any object that moves possesses energy. Energy does not have mass, does not take up space, and cannot be seen, heard, felt, tasted, or smelled. We experience the effects of the energy on matter, not the energy itself. For example, you feel heat or an electrical shock because energy causes changes in the matter making up your finger.

4.1 Work When energy acts upon matter, it does so in the form of a force. A force is any influence that changes the speed of a body of matter, its direction of motion, or both. The quantity called work is a measure of the amount of change that a force brings about when it acts upon something. Work is defined as the amount of force exerted onto an object multiplied by the distance the object moves in the direction of the force. The relationship between work, force, and distance is:

W = d(F cos ) where W is the work, d is the distance the object

moves, F is the force, and is the angle between the displacement and force applied. When the action force is measured in newtons and the distance is measured in meters, then

the unit of work is the newtonmeter, also called the joule (J). Two things happen when work is done: (1) a force is exerted and (2) something is moved in the direction of that force. However, a force does not always cause something to move. If a force does not cause something to move in the direction of the force, no work is done on it.

Work is a scalar quantity. Although work itself has no direction, the direction of the force and motion vectors determine how much work is done on an object. When a component of a force acts in the same direction as the motion of the object, work is being done. When the direction of the force is perpendicular to the motion of the object, no work is done.

4.2 Power Like the term work, the term power has both an everyday meaning and a scientific meaning. When we speak about power in everyday life, we generally refer to great strength or authority. The scientific meaning of power, however, is the speed, or rate, of doing work. In other words, power is the amount of work performed in a given time period. In the metric system, power is expressed in watts. One newton-meter per second is equal to one watt. Like the term work, the term power has both an everyday meaning and a scientific meaning. When we speak about power in everyday life, we generally refer to great strength or authority. The scientific meaning of power, however, is the speed, or rate, of doing work. From a practical standpoint, the fact that a certain amount of work is done is not always good enough. The question of how long it takes to do this work often determines the practical values of certain devices. The quantity we use to measure how rapidly work is done is called power. The unit of power is joules per second

Figure 4-1: Work is performed when a force acts to move an object.


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which is given the name watt (W). Common formulas used to calculate the power of a device or object are:

v F t

Fd

t

W P

where P is the power of the device.

4.3 Conservation of Mechanical Energy Mechanical energy is the energy with which moving objects perform work. There are two basic states of energy: kinetic and potential. (1) Kinetic energy is energy that an object has when it Is moving. Examples include:

a rock falling off a cliff.

heat given off by a burning lump of coal. (2) Potential energy is stored energy that an object has because of its position or its chemical composition. Examples include:

a rock on a cliff (due to its position).

a lump of coal (due to its chemical makeup).

In 1807, the English scientist Joseph Young suggested that this stored work be called energy, from Greek words meaning “work within.” The term energy is now widely used to describe anything that can be converted to work. In other words, energy is what something has if it can do work. When we say something has energy, we mean it is able to do work. The more energy something has, the more work it can do. On the other hand, when we do work on

something, we have added to it an amount of energy equal to the work done. In a sense, when work is being done, stored work, or energy, is being transferred from one thing to another. Potential energy may be changed into kinetic energy when motion is produced. Kinetic energy may also be changed into potential energy. Both potential and kinetic energy exist in many forms. Under ideal conditions (that is, in the absence of friction), this transformation occurs without any loss. Research establishing the relationship between work and energy was done in the nineteenth century by James Prescott Joule, for whom the unit of energy is named. The work-energy theorem is valid provided that no energy is lost to the surroundings through external forces. When the kinetic energy of the object increases, and work is done on the object, by convention, the work is considered to be positive. When the kinetic energy of the object decreases, and the object does the work, by convention, the work is considered to be negative. The law of conservation of energy states that energy cannot be created or destroyed. It may be transformed from one form to one or more other forms of energy, but the total amount of energy never changes. The transformation of energy from one form to another is an important part of this law. If we look at only one form of energy, it does not always seem to be conserved. In fact, most activities of everyday life involve one form of energy being transformed into another. In every transformation, though, some of the energy is converted into heat. The amount of energy a system transforms into a desired form of energy compared with the total amount of energy put into the system is known as the system’s efficiency. A common relationship to express these ideas is:

PE KE W where KE is kinetic energy and PE is potential energy. All change involves a flow of energy from one part of the environment (that loses energy) to

Figure 4-2: Examples of mechanical energies.


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Figure 4-4: Gravitational potential energy is based on the difference in height from the ground. In this case, the object on the left has more potential energy since it is higher up.

another part (that gains it). Although change occurs continuously throughout the environment, certain general characteristics tend to remain constant. This is the result of a natural balance among all the changes taking place, called environmental equilibrium. This equilibrium is easily upset on a small scale, for example by the burrowing of a worm in the soil, but it is normally not upset on a large scale. Human activities, however, often severely disrupt the environmental equilibrium. Environmental equilibrium resulting from opposing forces or actions balancing out is called a dynamic equilibrium. An example would be the level of a lake remaining the same even though thousands of liters of water move in and out of the lake per day. When environmental equilibrium results from little noticeable change it is static equilibrium.

4.4 Kinetic Energy The first phenomenon that was clearly recognized as energy was motion itself. Work involves motion since an object has to be moved through a distance. So, it was not surprising that motion could do work. Any object that moves contains energy because if it collides with another object, it can set that object’s mass into motion. Thus, it will do work on that object; it will exert a force on that object that will move its mass through a distance. The energy of motion is called kinetic energy, from a Greek word meaning “motion.” The faster an object moves, the more kinetic energy it has. The formula for kinetic energy is:

KE = ½mv2 where m is the mass of the moving object and v is its speed.

4.5 Gravitational Potential Energy Gravitational potential energy is the stored energy in a system resulting from the gravitational force between Earth and the object. The change in gravitational potential energy varies directly with the change in distance as compared to Earth’s surface. The height is one of the factors used in determining the amount of potential energy stored in the object. A second factor used in determining potential energy is the weight of the object. The potential energy of an object can be determined by using one of the following formulas:

PEg = mgh = Fgh where PEg is gravitational potential energy (often denoted only as PE), m is the mass of the object, g is the acceleration due to gravity, and

h is the change in height between the object’s position and Earth’s surface.

4.6 Elastic Potential Energy Objects that can return to their original form after being deformed are said to be elastic. A stretched rubber band possesses the ability to do work that does not depend on a gravitational field. A compressed spring also has the ability to do work. The energy stored in a rubber band or a spring is examples of elastic potential energy. The formula is:

PEs = ½kx2

Figure 4-3: Kinetic energy is based on both mass and speed.


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100 g

100 g

100 g

Figure 4-5: Elastic potential energy in a compressed spring is converted to kinetic energy when expanded.

where PEs is elastic potential energy (again usually denoted as PE), k is the elastic constant of the object (often called a spring constant), and x is the change in the object’s position from it rest point or equilibrium. Springs can be useful when they are compressed, stretched, or both. The force exerted by a spring is zero when the spring is in its relaxed normal position. When the spring is stretched, the spring exerts a force that opposes the motion. Furthermore, the magnitude of the restoring force, as it is called, is directly proportional to how much the spring is stretched. This relationship is called Hooke’s law and can be calculated using:

Fs = – kx where Fs is the restoring force of the spring, k is the elastic constant, and x is the distance the object has been stretched or compressed.

At times, the amount of time needed for the spring to oscillate (bounce back and forth) is requested. If this is the case, the following equation is used:

T = 2 k

m

where T is the period (time to complete one oscillation), m is the mass attached to the spring, and k is the elastic constant.

The period of the pendulum (time to go once

over and back) depends only on the length of

the pendulum and g (the acceleration due to

gravity) for any mass and reasonably small

height of swing. On the surface Earth where g is

constant, only increasing the length of the

pendulum will lengthen the period:

T = 2 g

l

where T is the period, l is the length of the pendulum, and g is the acceleration due to gravity.

4.7 Internal Energy The internal energy of an object is defined as the sum of the energy of all its molecules, counting all forms of energy except those the object has as a whole. If the whole object is moving, carrying all its molecules along with it, or if the whole object is raised, lifting all its molecules with it, we do not include this kinetic energy or potential energy of the molecules in the internal energy of the object. To be included in internal energy we must look internally, inside the object, to find the energy. Although each molecule’s mass is exceedingly small, their velocities are substantial; the forces they exert on each other are strong, and there are very many of them in even the smallest of objects. If we were to add the energy of all the

Figure 4-6: Energy relationships using a swinging pendulum.


38

Figure 4-7: Heat flows from higher temperatures to lower ones.

molecules contained in an object, the total amount would be significant indeed. Mechanical energy is converted into internal energy when work is done against friction. The increase in internal energy in turn raises the average kinetic energy of the molecules. As a result, whenever we rub our hands together, or drive a nail into wood, or drag an object over a rough surface, we can detect an increase in the temperature of these objects. The atoms and molecules of any object, be it in the solid, liquid, or gaseous state, possess various forms of energy.

Molecules move from place to place, particularly in the gaseous state. This provides each of them with translational kinetic energy.

Molecules rotate in place. This provides each of them with rotational kinetic energy. This form of energy is more prominent in the case of diatomic molecules than in the case of monatomic molecules; because diatomic molecules are longer (the KE of rotation depends on the radius of the circle described by the rotating object).

Molecules consisting of two or more atoms contain vibrational kinetic energy. A collision with another molecule may pull the atoms slightly apart, whereupon they respond like a stretched spring. They bounce back toward each other, then move apart again, come together again, rebound again, and so on.

Molecules have potential energy due to the attraction they exert on each other, just as Earth’s attraction on objects produces gravitational potential energy. The attractive forces between molecules are frequently much stronger than those exerted by gravity; these forces are electromagnetic in nature. This is evident from the fact that it is sometimes very difficult to pull molecules apart.

The amount of internal energy an object has depends on its temperature, mass, phase, and intermolecular bonds. One formula used to calculate internal energy is:

U = Q + W

where U is internal energy, Q is heat (either added to or removed from the object), and W is work.

4.8 Heat and Temperature The internal energy of an object can be changed either by changing the kinetic energy of its molecules, or by changing the potential energy of its molecules, or both. It is important to understand the difference between internal energy, temperature, and heat. Heat is a form of energy that flows between two systems in thermal contact. It is a scalar quantity and is measured in joules. Heat does not always flow between systems that are in thermal contact. The property of systems that determines whether or not heat flow will occur is called the temperature. If there is a temperature difference between two systems that are in thermal contact heat will flow from the system with higher temperature to the system with lower temperature. Two systems are said to be in thermal equilibrium if, when brought into thermal contact, no heat transfer occurs. In this case the systems must have equal temperatures. Whenever this occurs, the internal energy of both objects is changed.

Temperature is the measure of the internal energy of molecules; it is a fundamental quantity, not defined in terms of length, mass or time. It is the property of systems that determines the existence and direction of the heat flow between them when they are in thermal contact. It originates from the motions


39

Figure 4-8: Three commonly used temperature scales.

and vibrations of the molecules in matter. Temperature is not a form of energy. It is a measure of the average kinetic energy of the particles in a substance. When the average kinetic energy of the molecules increases, the temperature of the object increases. Although the Arctic Ocean is at a much lower temperature than a cup of boiling water, it has far more internal energy and can release far more heat. This is because temperature measures the average kinetic energy of the molecules, while internal energy is the total kinetic and potential energy of the molecules. Since the Arctic Ocean has so many more molecules than a cup of boiling water, the total kinetic and potential energy of its molecules is much greater.

4.9 Temperature Scales The most common perception of temperature is in terms of the hotness and coldness of an object. This is unsatisfactory for physics, because it involves a subjective and not that reliable. Definitions of quantities in physics are expressed in terms of physical parameters that can be measured. This is accomplished by using the height of a liquid, such as mercury or alcohol, in a glass tube. The higher the temperature, the higher the liquid level in the glass tube. The principle that the liquid expands and contracts the same amount for each degree of temperature change is used. If marks are placed on the tube for particular reference temperatures that can be reliably reproduced in advance, a thermometer is created. The temperature of the object is the same as that of the thermometer when thermal equilibrium is achieved. Three primary temperature scales in common use are the Celsius, Fahrenheit, and Kelvin scales. Each scale is based on different choices for setting values for two convenient fixed points: the freezing and boiling of water. Converting between the Celsius and Fahrenheit scales can be accomplished using the following formulas:

32 T5

9 T CF

32 T 9

5 T FC

where TF is the temperature on the Fahrenheit scale and TC is the temperature on the Celsius scale. The Kelvin scale is based on the existence of a lowest temperature below which (even to which) it is physically impossible to cool any system. It is designed to have its zero point represent the temperature at which an ideal gas would have zero pressure. This temperature is called absolute zero. There is an easy conversion between the Kelvin and Celsius scales:

TK = TC + 273.15 where TK is the temperature on the Kelvin scale.

The following should be observed with regard to quoting temperatures and temperature differences:

When quoting a temperature in Celsius or Fahrenheit the degree symbol is both written and spoken first. That is, 10oC ("10 degrees Celsius") for example.

When quoting temperature differences in Celsius or Fahrenheit the degree symbol is both written and spoken last. That is, 10 Co ("10 Celsius degrees") for example.

The Celsius scale is no longer alternatively referred to as the Centigrade scale.


40

No degree symbol is written or spoken with the Kelvin scale. That is, both temperatures and temperature differences of 10 are denoted by 10 K ("10 Kelvin").

4.10 Heat Capacity and Heat Transfer Heat flow is associated with a temperature difference between two systems. It is also true that heat flow can result in the change in temperature of a given system. The amount of heat needed to change the temperature of an object is directly proportional to the change in temperature required. The heat capacity of an object is the ratio of thermal energy added to or subtracted from the object and the corresponding change in temperature. The SI unit of heat capacity is J/K (or J/Co). The more massive a body is, the greater its heat capacity. The heat capacity also depends on the composition of the object. The concept of heat capacity refers to specific objects since it is often necessary to know how much of the substance needs to be heated or cooled (and the heat capacity contains that information). It is often more useful to have a quantity that is independent of the amount of substance you have and depends only on the nature of the substance. Such a quantity is called the specific heat of an object, which is basically the heat capacity per unit mass of the substance. The SI unit of specific heat is

J/(kgK) or J/(kgoC). Therefore, in terms of specific heat, the amount of heat needed to change the temperature of a substance is given by the following equation:

Q = mcT where Q is the heat required, m is the mass (in kilograms), c is the specific heat of the material,

and T is the change in temperature (in Kelvin or Celsius only). In a closed system, the amount of heat that leaves a hotter object is equal to the amount of heat that enters the cooler object when the two objects are in thermal contact. This is an important fact since an equation describing conservation of thermal energy, or conservation of heat, between objects can be written.

It has already been noted that heat is a transfer of energy and that the transfer of energy often comes about as a result of mechanical work being done. It follows that heat flow can either be converted into, or result from, mechanical work. Specialized units have been adopted for dealing with the mechanical work associated with heat flow. One unit of heat in common use is the calorie (cal). One calorie is the amount of heat needed to raise the temperature of 1 gram of water by 1 Celsius degree. The equivalent amount of energy in joules is called the mechanical equivalent of heat:

1 cal = 4.186 J The amount of temperature change that objects experience whenever heat is transferred depends upon the specific heat capacity of the objects. The standard for measuring specific heat capacity is the behavior of water - the amount of heat that is required to raise the temperature of one gram of water by one degree Celsius. Unfortunately this unit was established in the early days of the investigations of thermodynamics before the equivalence of the various forms of energy was established in physics. Heat can be measured in joules as other forms of energy are measured, but the use of the calorie as a unit is firmly established historically. The term calorie used in reference to food and diets is actually 1000 times as large as the physics calorie. The dietary term is usually referred to as the Calorie with a capital letter. That is to say that 1 Calorie as used by a nutritionist is the amount of heat necessary to raise the temperature of 1000 grams (1 kilogram) of water by 1o C. Of the three methods of heat transfer, the most familiar is conduction; the process whereby heat is transferred because of direct contact between objects. Heat conduction occurs when heat flows directly through an object because of a temperature difference across the material. The average rate at which heat will flow, Q/t (where t is time), is found to depend on three clearly identifiable quantities:

the temperature difference.

the area through which it flows .


41

Figure 4-9: Methods of heat transfer.

Figure 4-10: An exaggerated representation of thermal expansion.

the distance through which it flows. A material that easily allows such heat transfer, for example a metal, is called a good conductor while a material that does not allow such heat transfer, for example Styrofoam, is called an insulator. Heat transfer also occurs by the movement of matter from one place to another such as hot air rising in a room. This type of heat transfer is called convection. Convection is an important process in fluids where material is free to move. Heat transfer by radiation occurs even when there is no medium present and is responsible for all the energy the Earth receives from the Sun. All objects emit and absorb heat in the form of radiation (electromagnetic waves). There are many forms of electromagnetic radiation; infrared and visible light are the two that are most applicable to heat transfer.

It has been determined that the radiant power of an object is dependent on the surface area of the object from which the radiation flows and the temperature, and the emissivity which depends on the nature of the surface. The emissivity is a dimensionless quantity whose value lies between 0 and 1; it is a measure of how effectively an object radiates heat. If emissivity equals 1 the object is said to be a perfect radiator. Objects also absorb radiant energy according to this relationship except that the temperature is the temperature of the environment instead of the temperature of the object. An object that is black in color is both a better radiator of energy and a better absorber of energy.

4.11 Thermal Expansion Most substances expand when heated and contract when cooled. A liquid-filled thermometer is a great example of thermal expansion and contraction of liquids although any phase may experience the process. The amount of expansion is different for different substances. Some aspects of this behavior can be identified that are the same for nearly all substances. When a substance is heated, it expands not only in length but also in width and thickness. Therefore, thermal expansion results in a change in length, area, and volume. It is found that the amount that a substance will expand is directly proportional to the temperature change that drives the expansion as well as the original size of the object being heated or cooled. The SI unit for thermal expansion coefficients is K-1 or (Co)-1.

The coefficient of linear expansion tells what change in length takes place per unit length when the temperature of the substance goes up 1 degree. For example, the coefficient for brass is 1.9 x 10-5 per °C (°C-1). When a substance is cooled the corresponding contraction takes place and the same thermal coefficient can be used. The following formula can be used to calculate linear expansion:

l = liT


42

where l is the change in length experienced by

the object, li is the original length of the object,

is the coefficient of linear expansion, and T is the temperature change. For the expansion of areas, the description is very much like that of linear expansion. To a very high approximation, the coefficient of areal expansion works out to be twice that of the coefficient of linear expansion. The following formula results:

A = (2)Ai T

where A is the change in area and Ai is the original area of the object. The coefficient of volumetric expansion tells what change in volume takes place per unit volume when the temperature goes up 1 degree. For example, when the mercury thermometer is heated, the mercury expands more than the glass bulb. Therefore the level of the mercury rises to compensate for the difference in volume expansion. Water behaves peculiarly in this respect. As water is cooled from 100°C it contracts until its temperature reaches 4°C. If it is cooled further the water will expand; that is, water is densest at 4°C. The following equation can be used for volume expansion:

V = ViT

where V is the change in volume experienced

by the material, Vi is its original volume, and is the coefficient of volumetric expansion. NOTE: A good approximation for the coefficient of volumetric expansion of most substances is three times that of the coefficient of linear expansion.

4.12 Phase Changes Matter exists in nature in three phases - solid, liquid, and gas - under ordinary conditions. Heat flow is required to change the phase of a substance although a change in pressure can also cause a change from one phase to another without changing its chemical identity. When matter is transformed from one phase to another, there is no change in temperature. The internal energy of the substance changes;

molecules in the liquid phase have a higher average energy than they do in the solid phase, and they have an even higher energy in the gaseous phase. This energy is transferred at the melting point for the transfer from solid to liquid and at the boiling point for the transfer from liquid to gas. The amount of heat that is required to completely convert one kilogram of a substance from one phase to another is called the latent heat (L). The SI unit of latent heat is J/kg. The value of L depends on the type of phase change being considered. The boiling of a liquid occurs at the temperature at which the equilibrium vapor pressure equals the external pressure on the liquid. Steam condensing on your hand produces severe burns due to the heat liberated on condensation. Steam at 100oC is much more damaging than water at the same temperature. The particles that escape from the surface of a liquid are those having the highest kinetic energy. The remaining particles have lower average kinetic energy. Thus, heat must be added to the liquid to maintain constant temperature. Vaporization is therefore an endothermic process – requires the addition of heat. The heat of vaporization of a liquid is the number of joules per gram of liquid that must be added in order to maintain a constant temperature while vaporization, or boiling, occurs. Heat of condensation is the number of joules per gram that must be removed in order to maintain constant temperature during condensation – the change of a substance from a gas to a liquid. The process of condensation is exothermic – requiring the removal of heat. The number of joules for both the heat of vaporization and heat of condensation are equal in magnitude and can be found using the following formula:

Q = mLv where Lv is the latent heat of vaporization. When a liquid is cooled (heat is removed), a temperature is eventually reached at which the liquid begins to freeze. It changes to a solid. This temperature, which remains constant until all the liquid has solidified at SAP, is called the freezing point of the liquid. While the liquid is cooling, the average kinetic energy of its


43

Figure 4-11: A typical heating curve.

particles decreases until it is low enough for the attractive forces to be able to hold the particles in the fixed positions characteristic of the solid phase. Alternately, warming a solid eventually causes the solid to melt. It begins to change to a liquid. During this phase change, the temperature remains constant until the entire solid has liquefied at SAP. This temperature is called the melting point of the solid. For any given substance, the melting point temperature is exactly the same as the freezing point temperature. The only difference is in the direction of approach. The melting-freezing point of a substance may also be defined as the temperature at which the liquid phase and the solid phase exist in phase equilibrium. Melting is an endothermic process. In order to maintain a constant temperature during this phase change, heat must be continually added to the system. The amount of heat needed to change a unit mass of a substance from solid to liquid at constant temperature and 1 atmosphere of pressure is called the heat of fusion of that substance. Due to the principle of the conservation of energy, the amount of energy to refreeze the water, heat of solidification, is equal to the heat of fusion and can be found using the following formula.

Q = mLf where Lf is the latent heat of fusion. Under certain conditions, it is possible for a substance to change from a solid directly into the gas phase without obviously passing through the liquid phase. This solid-to-gas change is called sublimation. Iodine and carbon dioxide are examples of substances that may undergo sublimation. Iodine and carbon dioxide have small non-polar molecules with weak intermolecular attractions. The reverse of sublimation is called deposition.

4.13 Ideal Gases Since so many gases exhibit the same behavior, a model was developed to help explain this similarity. This kinetic molecular theory of gases is based on four assumptions. Gases that have these properties are known as ideal gases.

All gases are composed of tiny, individual particles called molecules that are in continuous motion. These particles move rapidly, randomly and in straight lines.

When particles collide with one another, energy is transferred without loss, from one particle to the other. Therefore, the net total energy of the system remains constant.

Compared to the distances between them, the particles are so small that their volumes are considered to be zero.

The particles have no attraction for one another.

Real gases do not behave exactly as predicted by the “ideal” gas model. Deviations from the ideal behavior are due to the fact that gas particles do have volume and they do exert some attraction for one another. Deviations from are least obvious among light gases at high temperatures and low pressures. These conditions are optimum for high kinetic energies and maximum separation of particles. Hydrogen and helium are closest to “ideal” gases. Gas volumes are influenced considerably by changes in temperature and pressure. Thus, when working with gases, it is convenient to define standard reference conditions. By convention, 273 K and 1.01 x 105 Pa represent


44

standard temperature and pressure, often designated as STP. When dealing with a gas volume, STP is assumed unless otherwise indicated. An equation of state shows how the pressure, temperature, number of particles, and volume of the gas depend on each other is:

PV = NkT where P is the pressure of the gas, V is its volume, N is the number of particles, k is the Boltzmann constant with a value of 1.38 x 10-23 J/K, and T is the temperature of the gas in kelvin. An alternative way to write the equation of state for an ideal gas uses the concept of the mole. A mole is the amount of a substance that contains 6.02 x 1023 entities; this value is called Avogadro's number. This leads to a commonly used alternative version of the equation of state for an ideal gas, called the ideal gas law:

PV = nRT where n is the number of moles in the gas and R is the “universal” gas constant with a value of

8.31 J/(mol K). The thermal energy of gas molecules is due to their random motion. The temperature is their average kinetic energy that depends on both their mass and speed. If a constant amount of a gas is heated, its mass cannot change so the

molecules must go faster to be at a higher temperature. If they go faster, they should collide harder and more frequently on the sides of the container. For a constant volume of gas, higher temperatures lead to higher pressures. If the pressure were to stay constant when the temperature increases, the volume must become larger so that faster moving molecules are farther apart. Even though the molecules at higher temperature are moving faster, their lower concentration compensates so the pressure can stay constant. As a result, as the temperature of a gas increases, for it to remain at constant pressure, its volume must increase. At constant temperature if the volume of a gas is decreased, the concentration of molecules increases making more frequent collisions. As a result, as the volume decreases the pressure increases. The properties can be combined into one equation known as the combined gas law for an ideal gas:

2

22

1

11

T

VP

T

VP

where temperature must be in Kelvin. NOTE: If certain variables are held constant, the

general formula is manipulated and referred to by a specific gas law name, such as Boyle’s Law or Charles’ Law.

The advantages of using the combined gas law are:

only one equation is needed

all initial values are on the left of the equation; all new values are on the right of the equation.

in cases where either temperature or pressure is unchanged, it can be ignored (or included); its value will not affect the problems

4.14 Thermodynamics Thermodynamics is the study of the relationships among heat, work, and energy in the universe. These conversions often are considered at the particle level. The principles of

Figure 4-12: The ideal gas law.


45

Figure 4-14: Plot of pressure vs. volume for an isobaric process.

thermodynamics are based on our experiences in observing nature. Thermodynamics has many applications in disciplines ranging from physics and engineering to biology and medicine. The laws of thermodynamics are based on the law of conservation of energy and can be summarized in three fundamental laws. The First Law The first law of thermodynamics is a restatement of the law of conservation of energy. It states that the change in the internal energy of a system is equal to the heat that the system absorbs (or releases) minus the work it does (or has done on it). In symbolic form the first law is written as follows:

U = Q + W Although energy and work are positive quantities, energy can be added to or subtracted from the system and the work can be done to or by the system. The volume, pressure, and temperature of a gas can be changed in four different ways:

compressing the gas; a.k.a. work done on the system (positive W)

letting the gas expand; a.k.a. work done by the system (negative W)

adding heat to the gas (positive Q)

removing heat from the gas (negative Q).

It is important to use signs correctly. Changes in internal energy will be accordingly positive or negative. For a gas that expands or contracts during a process while held at constant pressure (known as an isobaric process), the work done by the gas during the process is found to be:

W = P(V)

The result can be interpreted graphically as the area under the curve of a pressure versus volume plot. The area under the curve equals the work done by (or on) the gas for any process.

At constant volume (called an isochoric or isovolumetric process), no work is done by the gas during a reversible process. This fact is consistent with the above mathematical

expression because V = 0 when the volume is held constant. Also, there should be no work done because work results from force acting through distance and if the gas does not expand or contract through any distance you would expect the net work done by the gas to be zero.

An isothermal process is one that takes place at constant temperature. For an ideal gas, the relationship between the pressure and the volume during an isothermal process is PV =

Figure 4-13: Heat added to a gas in a closed container will cause the gas to expand and move the “lid” of a distance while the gas expands.

Figure 4-15: Pressure-volume graph for an isochoric process.


46

Figure 4-16: Pressure-volume graph for an isothermal process and an adiabatic process.

Figure 4-17: Second law of thermodynamics – entropy.

NkT. Work is done by the gas during an isothermal process. If no heat flows into or out of a system during a process the process is called adiabatic. This type of process occurs when a system is well insulated or when the process takes place so rapidly that heat doesn't have time to flow. During adiabatic processes, the pressure, volume, and temperature may all change.

The Second Law The second law is a result of the work of the French physicist Nicolas Carnot with heat engines. A heat engine is device that converts thermal energy or heat to mechanical work. This law states that heat cannot flow from a colder object to a warmer one without work being done on the system. Another consequence of the second law is that heat can never be converted completely into work. In other words, no heat engine can be 100 percent efficient. Some of the heat absorbed by the engine must be lost in the random motions of its molecules. The quantity known as entropy (S) is a measure of this disorder. The statement

that the entropy of the universe increases is yet another equivalent way to state the second law of thermodynamics. In fact the second law of thermodynamics is commonly called "the law of entropy." The amount of work done by a heat engine can be found by using:

W = Qh – Qc where Qh is the heat added to the system and Qc is the energy removed from the system. To calculate the efficiency of a machine, one of the following formulas can be used:

e = hQ

W

e =h

c

Q

Q - 1

where e is the efficiency of the machine. The maximum possible efficiency of a heat engine that operates from a single hot reservoir and a single cold reservoir results when all processes in the cycle are reversible - this statement is known as Carnot's theorem. This leads to an equation for maximum efficiency to be dependent upon the temperatures of the reservoirs and not on any details of engine:

e = h

c

T

T - 1

where Tc is the temperature of the cold reservoir in kelvin and Th is the temperature of the hot reservoir also in kelvin. It is important to remember that the descriptions of heat engines refer to devices that operate in cycles, meaning the devices return to some initial state after going through some processes and then repeat the cycle. This means the internal energy of the system is the same at the end of a cycle as it was at the beginning of the cycle. Refrigerators, air conditioners, and heat pumps use work to generate a flow of heat against its


47

natural tendency. A refrigerator is almost the precise reverse of a heat engine. In a refrigerator work is the input that forces heat to flow from a cooler region (the refrigerator) to a warmer region (the room). An air conditioner is the same as a refrigerator except that the room being cooled is the cold reservoir and the outdoor air is the warm region. A heat pump is the reverse of an air conditioner. Work is consumed to remove heat from the cold reservoir of outdoor air and pump it into the warm reservoir of the room being heated. In an ideal heat pump the amount of work required to pump an amount of heat into a room is given by the same relationship as for a Carnot heat engine.

The Third Law The third law of thermodynamics states that it is impossible to reduce the temperature of a system to absolute zero. In order to lower the temperature of a system, heat must be removed from the system. One way to remove heat from a system is to place it in contact with another system at a lower temperature. In order to lower the temperature of a system to absolute zero, it would have to be placed in contact with a system whose temperature was lower than absolute zero. Since absolute zero is time lowest possible temperature, this is impossible. Another way to remove heat from a system is to allow the system to do work. This method can be used to lower the temperature of gases. Below 3.2 K all substances are in the liquid or solid phase, so absolute zero cannot be achieved in this way either. Scientists have been

able to reach temperatures lower than 1 K by pumping away the vapor emitted by supercold liquids and by using a variety of magnetic effects. However, the closer to 0 K a system gets, the more difficult it is to further reduce its temperature. Absolute zero has never been achieved and is assumed to be unattainable. Quickly summarizing some of these results:

Heat gained is equal to heat lost.

Heat never flows from a cooler body to a hotter body of its own accord.

Heat can never be taken from a reservoir without something else happening to the system.

Entropy is a measure of the random disorder in a system.

The entropy of the universe is increasing.

Figure 4-18: A refrigeration system is pictured on the left and a heating system on the right.

Physics Honors Fluid Mechanics

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Chapter 5 – Fluid Mechanics

Solids resting on a surface press down on the surface with a force equal to its own weight. Fluids are characterized by their ability to flow; both liquids and gases are considered to be fluids. Fluids have weight and therefore exert forces on other objects by pressing on them. Fluids vary in the amount of force they can exert because of differences in their density.

5.1 Density One of the more convenient properties used to describe a substance is its density. The density of a substance is a measure of how compact the substance is; how much mass is packed into a volume of the substance. The formula used for density is:

V

m

where is density, m is mass, and V is volume. Under ordinary conditions, the density of a substance is a constant value. Since changes in temperature and pressure affect the volume of liquids and gases, their density is dependent on these factors. The density of water at standard temperature and pressure (STP) is 1 g/mL or 1 kg/L. Since the basic unit for density in the SI system is kilogram/cubic meter, the generally used as a standard for comparison is given as 1.00 x 103 kg/m3. Substances of greater density exert more force on a particular surface than those of lesser density.

A way to compare densities is called specific gravity which is often abbreviated SG. The specific gravity of a substance is the ratio of its density to the density of water. Basically, if an object’s density is greater than the density of water the object will sink. If the object’s density is less than that of water, it will float. Specific gravity is a dimensionless quantity that is found using:

w

o

where o is the density of the object and w is the density of water, 1.00 x 103 kg/m3.

5.2 Archimedes' Principle and Buoyancy When an object is submerged in a fluid, the object’s volume displaces an equal volume of the fluid. The pressure applied by the fluid onto the object results in an upward force on the object known as buoyancy or a buoyant force (FB). This phenomenon is governed by Archimedes' principle which states that an object immersed in a fluid experiences an upward force equal to the weight of the fluid displaced by the object. The weight of the fluid displaced by the object equals the mass of the fluid times the acceleration due to gravity. When dealing with buoyancy it is usually more convenient to express the fluid’s mass in terms of its density

Figure 5-1: Archimedes’ principle at work. Adapted from: www.exploration21.com

Negative Buoyancy

If an object’s weight is MORE than the buoyant force (weight of displaced fluid), the object SINKS.

Neutral Buoyancy If an object’s weight is EQUAL to the buoyant force on the object, the object “HOVERS” or can be placed anywhere in the fluid.

Positive Buoyancy If an object’s weight is LESS than the buoyant force, the object will FLOAT.


49

when calculating its weight using the equation

Vg instead of mg as is normally used to calculate weight.

FB = fluidVobjectg

where FB is the buoyant force experienced, fluid is the density of the fluid in which the object is submerged, Vobject is the volume of the submerged portion of the object, and g is the acceleration due to gravity (a commonly accepted standard of 9.81 m/s2). The buoyant force applied to submerged objects opposes the downward force of gravity and makes it appear that the object loses weight while immersed in a fluid – called its apparent weight.

Fg = Fg – FB

where Fg is the object’s apparent weight, Fg is the object’s “actual” weight or more accurately its weight in air, and FB is the buoyant force on the submerged object. Archimedes' principle explains the phenomenon of floatation which occurs when the buoyant force acting on an object equals the object's weight. Many times objects are not completely submerged in the fluid; the principle holds for objects that are completely submerged in a fluid as well as for objects that are not. If only a portion of an object is submerged, the amount

that is submerged (Vsub) can be found using the following relationship:

Fosub V V o

where Vo is the volume of the object, o is the

density of the solid object, and F is the density of the fluid in which the object is placed. Many times the percentage of the object submerged is needed and can be found using:

100 x %F

osub

Changing the shape of an object may change its ability to float. This is the idea behind metal ships being able to float. A flat sheet of metal will sink in water but when shaped into a boat it will float. The reason is that the “boat shape” has a greater volume than the flat sheet and therefore displaces a volume of water equal to its weight. When an object is submerged in a fluid, the object’s volume displaces an equal volume of the fluid.

Figure 5-2: Apparent weight is less than weight in air.

3.0 N

0.5 N

Figure 5-3: Buoyant force on a partially submerged object.


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5.3 Pressure Fluids push against the container in which they are placed in all directions. The pressure exerted by any substance is the amount of force (perpendicular to the surface) exerted by that substance on a given area. The units of pressure are those of force and area; in SI units this is called the pascal (Pa). One pascal is equal to one newton per square meter (N/m2). One of the formulas used to calculate pressure is:

A

F P

where P is pressure, F is force, and A is area. Earth’s atmosphere exerts pressure, called air or atmospheric pressure and is measured with a barometer. One type of barometer consists of a glass tube about one meter long, sealed at one end and partly filled with mercury. At 0°C at sea level, the atmospheric pressure supports a column of mercury 760 mm high. In honor of Evangelista Torricelli, who invented the mercury barometer, mm of mercury (mmHg) is referred to as a torr. This value is also known as one atmosphere of pressure (atm) or standard pressure (SAP). It is important to recognize that the atmosphere is a constant presence in most experiments. The standard value of atmospheric pressure in SI is Patm = 1.01 x 105 Pa.

5.4 Static Fluids Static fluids are fluids that are relatively stationary, that is not flowing. Liquids have weight and therefore exert forces on other objects by pressing on them. You may be familiar with water pressure because of its effect on your ears when swimming under water. As you go deeper in the water, the pressure increases and may cause discomfort to your eardrums. Experimentation has shown that, at different depths, liquid pressure depends on two factors: (1) the height of the liquid and (2) the density of the liquid. The equation used to calculate static liquid pressure at a particular depth is:

P = Patm + gh

where P is the pressure experienced at a specific depth, Patm (sometimes Po) is the

pressure of the atmosphere, is the density of the fluid in which the object is submerged, g is the acceleration due to gravity (the commonly accepted standard of 9.81 m/s2), and h is the height of the fluid above the submerged object or its depth. The value used for Patm is usually that of standard air pressure but the term represents the atmospheric pressure at the time of the experiment. It is also important to note the Patm is only used when the system is exposed to the atmosphere otherwise the term drops out leaving the formula as:

P = gh One of the basic properties of static fluids that is very important to understanding fluid behavior known as Pascal's principle – which states that external pressure applied to an enclosed fluid is transmitted throughout the fluid. Pressure applied to a confined fluid is transmitted through the liquid without loss and acts perpendicularly on the surface of the container. Pascal's principle is crucial to understanding the hydraulic lift. This device uses fluid pressure to convert a small input force into a large output force. The input force is applied to a fluid over a small area causing a pressure change that is transmitted to a larger area. The resulting force is larger and is able to support the large weight. Without any external pressure applied to a fluid, the pressure at a certain depth beneath the surface is related to the weight of the fluid above it. If there is external pressure exerted on the fluid, the pressure is transmitted to every point in the fluid and it must be added to the pressure due to the weight of the fluid. Mathematically since the hydraulic lift is a closed system, you can compare the pressure exerted on the smaller piston to the pressure at the larger piston using the following ratio:

2

2

1

1

A

F

A

F


51

where 1 and 2 represent the smaller and larger piston areas, respectively. Since a larger output force is the result of this process, the distance through which this larger force can move an object, d2, is smaller than the distance at the input, d1. The same volume of fluid moves at both the input and output sites, therefore the areas can be compared using:

2

112

A

Ad d

The characteristics of fluid pressure are:

the pressure exerted by a fluid is equally applied in all directions and depends on depth.

fluids seeks their own level.

the pressure is independent of the shape and size of the container.

pressure can be transferred throughout a fluid.

5.5 Fluid Dynamics Fluid motion is usually very complicated. However, by assuming ideal fluids useful models can be developed. An ideal fluid is:

incompressible – the density is constant

non-rotational – the flow is smooth, no turbulence

non-viscous – fluid has no internal friction

steady flow – the velocity of the fluid at each point is constant in time

During the smooth flow of a constrained fluid, it is assumed that the same amount of mass passes through each cross section of pipe in a given amount of time. This smooth flow condition leads to what is known as the continuity equation is a reworking of conservation of mass. It says that mass m1 flowing through an area A1 in a given time equals the mass m2 flowing through area A2 in that same amount of time. The amount of mass per unit time of a fluid of density flowing through

area A at speed v is Av. Therefore, the equation of continuity is

(Av)1 = (Av)2 For ideal fluids, (Av)1 = (Av)2. The quantity Av equals the volume flow rate of the fluid; therefore for an ideal fluid the volume flow rate is constant.

In the eighteenth century it was observed that the pressure exerted by a fluid in motion is less than that the pressure exerted by the same fluid at rest. For a fluid flowing at a constant level, an increase in speed must be accompanied by a decrease in pressure and vice versa. This effect is called Bernoulli's principle and is important in understanding air flow in many applications including flight. The same concepts used to describe the movement of particles apply to fluid dynamics. With fluids it is more convenient to express these concepts in terms of density and pressure rather than mass and force. With a fluid the idea of a particle is replaced with a small region of the fluid called a fluid element. The fluid element is defined in terms of its density moving at speed, v, while covering a volume due to a

Figure 5-5: Fluid flow through a tube of differing areas.

Figure 5-4: The hydraulic lift is a practical application of Pascal’s principle – the pressure is equal at both pistons.


52

varying fluid pressure. Bernoulli's principle is a reworking of conservation of energy. The equation can be obtained by applying the work-energy theorem to the fluid element which leads to:

P1 + ½v12 + gy1 = P2 + ½v2

2 + gy2

where y1 and y2 refer to changes in vertical height – many times it is more logical to use the variable h to represent height. In summary:

based on the equation of continuity, the fluid pressure falls as the flow speed increases.

the fluid has different speeds and therefore different kinetic energies at different parts.

changes in energy must result from work being done on the fluid.

the only forces in a confined space the forces associated with changes in pressure from place to place.

Figure 5-6: Bernoulli’s principle.

Physics Honors Mechanical Waves

53

Chapter 6 – Mechanical Waves

A disturbance is a change in a body of matter. This change is opposed by a force of nature. The body of matter becomes the medium through which the disturbance travels. Traveling disturbances begin at a point of origin and transmit energy through the medium. A wave is a disturbance that carries energy through matter or space, without transferring matter. All waves are caused by vibrations. A single vibratory disturbance is called a pulse.

6.1 Types of Waves There are two main types of waves with which physics concerns itself: transverse and longitudinal. These types are defined by the relationship between the direction of the oscillation of the medium in which the wave travels and the direction of the transmission of the wave. In a transverse wave the direction of oscillation is perpendicular to the direction of movement. A wave on a string is a good example of a transverse wave. In a longitudinal wave the direction of oscillation is parallel to the direction of movement. Longitudinal waves are often referred to as compression waves since energy traveling the medium causes it to compress and expand such as in a spring.

6.2 Wave Characteristics Waves cause the particles of a medium to vibrate about their rest position; but it is the energy of the wave, not the particles of the medium, that travels through the medium. Periodic or continuous waves result from repeated disturbances within a medium. One complete repetition of the pattern in a periodic wave is referred to as a cycle. For example, a crest followed by a trough or a compression followed by a rarefaction, make up one cycle. A crest is the topmost point in a transverse wave which corresponds to an area of compression (high density and pressure) in a longitudinal wave. Likewise a trough, the lowest point in a transverse wave, corresponds to a rarefaction. One of these characteristics is the minimum time required to complete a cycle or repeat itself, called the period of the wave (T). The nature of the medium through which the wave travels does not affect the period. As with any simple harmonic motion the inverse of the period is called the frequency (f). The number of cycles or complete oscillations produced by a vibrating source per second is the frequency of the wave. It is also represents the number of cycles that pass by a fixed point per second. Cycles per second are expressed in the SI unit hertz (Hz), a derived unit equal to one s-1. Like period, frequency is determined by the source of the disturbance, not the medium. The frequency of a wave is the reciprocal of its period. The relationship between period and frequency can expressed as:

f

1 T

where T represents the wave period and f represents the wave frequency. Waves repeat themselves spatially. The length of one complete cycle is called the wavelength of the wave, symbolized by the Greek letter

lambda (). Wavelength can be measured from crest to crest, trough to trough, or between corresponding points on adjacent pulses. Both the source and the medium affect the wavelength.

wave direction

Figure 6-1: Typical waves: transverse (top) and longitudinal (bottom).

wave direction compression

rarefaction

disturbance direction


54

The velocity of a wave is determined by the properties of the medium and equals the distance the wave travels before it repeats divided by the time it takes to repeat. In order to transmit waves, the particles of the medium must have both inertia and elasticity that depend on its chemical composition and its physical characteristics, such as temperature and pressure. Common formulas for wave velocity are:

v = f

T v

where the v is the velocity or speed of the wave,

is the wavelength, and f and T are defined as before. The amplitude of a wave, usually represented by the letter A, is determined by its source. It is the maximum displacement of the wave from its equilibrium position and is a measure of how much energy a wave transports. The greater the amplitude of a wave, the more energy It carries. A mechanical wave transports energy in a medium through vibrations and can be either transverse or longitudinal. The vibrations travel at a definite speed, creating a regular motion in the medium without any net transfer of particles. In order to transmit waves, the particles of the medium must have both inertia and elasticity that depend on its chemical composition and its physical characteristics, such as temperature and pressure.

6.3 Propagation

Waves and pulses can be transmitted through different phases of matter. GASES When a tuning fork is struck, it oscillates back and forth. As one prong of the fork swings outward, it pushes air molecules in that same direction, creating a pocket of compressed air. As this compressed air expands and returns to normal, it pushes against the adjacent volume of air, compressing it forming a crest. The disturbance travels through the air (medium) away from its source. LIQUIDS If a pebble is dropped into a pond circular ripples are formed by disturbances traveling in the water. A crest is formed by the downward thrust of the pebble into the water. The forces of tension and gravity then pull this circular crest down. The crest collapses in one spot and reemerges farther away from the point of origin. This turns the ripple into an ever-expanding circle whose center is the point where the pebble entered the water. SOLIDS If a person holds one end of a rope that is attached to a door knob and moves his or her hand upward, a pulse in the form of a crest is formed in the rope. From its point of origin at one end, the crest moves toward the other end of the rope, transmitting the energy gained when

crest

trough

amplitude

wavelength

Figure 6-2: Common wave properties.

Figure 6-3: Compression and rarefaction in a gas.

Figure 6-4: Wave fronts in a liquid.


55

the person moved the rope. The force of tension in the rope acts to pull the crest down. As the crest comes down in response to these forces, it pushes against the adjacent part of the rope, causing the crest to reappear farther along the rope. This happens repeatedly as the crest travels the length of the rope. Energy traveling on a string or cord is a common way to generate a wave. The speed at which a wave on a string travels is determined by two properties of the string: its tension and its density. The more tightly pulled the string is, the more rapidly it will oscillate and the faster the wave will travel. The “heavier” a segment of the string is, the more slowly it will move under a given tension and the slower the wave will travel. If a wave was sent along a string with a fixed end, when the wave reaches the boundary it will be reflected back in the opposite direction along the string. Since the end is fixed it inverts each wave pulse upon reflection by applying a force in a direction opposite to the force applied by the string on the fixed connection. If the end of the string was free to move, the reflected wave

would not be inverted relative to the initial wave because the end oscillates along with the rest of the string.

6.4 Wave Phenomena Waves interact with each other and with boundaries and obstacles in a variety of ways. Wave phenomena include reflection, refraction, interference, and resonance. When a wave enters a new medium, reflection, transmission, and absorption can occur. Absorption occurs if some of the wave’s energy is converted to internal energy of the medium. There is always some absorption because the particles in a medium are not perfectly elastic. When a wave enters a new medium, it splits into a transmitted wave and a reflected wave. A transmitted wave is a wave that continues forward into the new medium. A wave that strikes a boundary between two media is referred to as an incident wave. A reflected wave is an upright or inverted returning wave that results from some of the energy of the incident wave’s pulse being reflected backward. When there is a great difference between the two media, the amplitude of the reflected wave is different from the amplitude of the transmitted wave. When there is very little difference between the two media, the reflected wave is much smaller than the transmitted wave. When a wave crosses a boundary between two different media, the properties of the media may result in a change of speed of the wave. If the incident wave does not approach the boundary at a right angle, the change in speed results in a change of direction of the wave. The change of direction of the waves at the boundary

Figure 6-5: The progressions of wave crests in a solid.

Figure 6-6: When a wave travels through matter of different densities, reflection and transmission are affected .

Figure 6-7: Wave inversion - when a wave meets a fixed boundary, the outgoing wave is inverted.


56

between two different media is called refraction. Since wavelength, frequency, and speed are interrelated, a change in speed must result in a change in at least one of the other two factors. If the frequency remains constant there must be a change in wavelength when the wave enters the second medium. Diffraction refers to various phenomena which occur when a wave encounters an obstacle. It can be the apparent bending of waves around small obstacles and the spreading out of waves past small openings. Two or more waves may pass through a medium at the same time. When this occurs two rules apply: (1) The total displacement experienced at any point where waves meet is equal to the sum of the displacements of the individual waves at that point. This is known as the principle of superposition. (2) Waves pass through each other, with each wave unaffected by the passage of the others. After meeting, the individual waves continue traveling in their original directions and with the same characteristics they had before they met. Superposition of two waves with the same frequency and amplitude traveling in opposite directions in a medium results in standing waves. They are most often produced by the reflection of a wave at a fixed boundary. A standing wave is a wave that oscillates in time but is fixed in its spatial location; a pattern of nodes and antinodes. A node (N) is the stationary point where two equal wave pulses meet and are in the same location, having a displacement of zero, whereas an antinode (A) is the point with the largest displacement. The nodes and antinodes in standing waves show

the effects of constructive and destructive interference. Points on a periodic wave that are identically displaced from the equilibrium position and are moving in the same direction away from the equilibrium position are said to be in phase. The phase difference between them is 0o. Points that are in phase are always a whole number of wavelengths apart. Points that do not meet all of these requirements are said to be out of phase. Wave interference patterns are regions where two or more waves are superimposed on each other. One of two general cases is when the individual waves add in such a way that their maxima (highest points) and/or their minima (lowest points) are at the same place at the same time. In other words, the waves are in phase. The result of this occurrence is that the amplitude of the resultant wave equals the sum of the amplitudes of the individual waves that met. The resultant wave is larger as compared to the individual waves. This effect is called constructive interference. Maximum constructive interference occurs at points where the phase difference between the waves that meet is 0o

.

Figure 6-8: When a wave passes through a barrier, it spreads out on the other side of the barrier.

Figure 6-9: A standing wave has one more node than antinode.

node

antinode

Figure 6-10: Points A, B, and C are in phase as well as points E and F. Points A and D are 90

o

out of phase, but points A and G are 180o

out of phase.

G

D

Equilibrium

A

B

C

E

F


57

Figure 6-13: The interference of a crest and a trough temporarily causes cancellation.

resultant amplitude = 0

A

A

destructive crest + trough

Figure 6-11: Constructive interference with equal amplitudes.

resultant amplitude = 2a

A A

constructive crest + crest

2A

resultant amplitude = 2a

A A

constructive trough + trough

2A

The other general case, destructive interference, occurs when the maximum of one wave meets the minimum of another. In this case the amplitude of the resultant wave equals the difference of the amplitudes of the individual waves. The resultant wave is smaller as compared to the individual waves. When destructive interference occurs the waves are said to be out of phase. Maximum destructive interference occurs where the waves are 180o out of phase. A special case of destructive interference is the total destruction of overlapping periodic waves if they have equal amplitudes and frequencies and are everywhere 180o out of phase.

Most objects have a natural frequency of vibration. If struck, they respond by vibrating at a particular frequency. When an object is disturbed by a wave whose frequency is the same as its natural vibration frequency, the amplitude of vibration of the object continues to increase. This is known as resonance; a special form of simple harmonic motion that occurs when small forces are applied at regular intervals to an oscillating or vibrating object. Tuning forks are designed to vibrate at a single frequency. When two tuning forks are placed near each other, the sound of one tuning fork creates small forces on the second tuning fork if the frequency of the second turning fork is the same as the frequency of the vibrating turning fork, resonance will make it vibrate. But, if the frequency is different from the frequency of the vibrating tuning fork, nothing will happen.

6.5 Sound Energy Sound is a form of mechanical energy produced by a vibrating object. When an object vibrates, it moves rapidly back and forth - this motion pushes and pulls the surrounding air, producing alternating compressed and expanded layers of air particles called sound waves. The compressions and rarefactions of sound waves consist of molecules vibrating parallel to the motion of the wave. Sound waves, therefore, are longitudinal waves. Sound waves spread outward in all directions from their source, somewhat like the circular ripples produced when a pebble is tossed into a pond. Sound is a mechanical wave that needs a medium in which to travel; it cannot travel in a vacuum. The substance that sound travels through is called its medium and be in any form of a solid, a liquid, or a gas. Figure 6-14: A vibrating object produces wave fronts

which are areas of compressed air that spread out from the source.

Figure 6-12: Constructive interference with unequal amplitudes. In all examples of interference, the combination is not permanent.


58

The speed of sound depends mostly on the density of the substance or medium, through which it is passing. The denser the medium is, the faster the sound waves can travel through it. Generally, sound travels fastest through solids and slowest through gases. Although the speed of sound can vary, it is always much slower than the speed of light. The speed of sound in air at standard temperature and pressure (STP) is 331 m/s. For each one degree Celsius change in temperature the speed of sound changes 0.60 m/s. For colder temperatures the speed of sound is less than 331 m/s and for warmer temperatures it is higher than 331 m/s. The speed of sound in air at room temperature (20oC or 68oF) is approximately 343 m/s. This is generally calculated using:

ΔT

Co

m/s 0.60 m/s 331

where T is difference in temperature from 0oC.

6.6 Sound Properties All of the generalized mechanical wave characteristics apply to sound waves. One of the most common sound wave properties experienced by humans is an echo. An echo is the reflection of a sound wave. Multiple echoes are called reverberations and can be useful in SONAR. The energy of a sound wave is determined by its source. As the amplitude of a sound wave increases, the loudness of the sound increases. The intensity of the sound is also related to the wave’s amplitude. As distance from the source increases, the intensity decreases. The sound wave’s frequency is used to describe the pitch, how high or low the sound is. A high frequency sound has a high pitch while a low frequency sound has a low pitch. The pitch of a sound source can change if the source is moving. The Doppler Effect refers to the change in frequency and wavelength caused by the movement of the source, the observer, or both the source and observer. When a source is moving toward you, the sound waves are

crowded together, producing a higher frequency and a higher pitch. When the source is moving away from you, the sound waves spread out, producing a lower frequency and a lower pitch.

6.7 Musical Instruments

All musical instruments – percussion, string, and wind – rely on vibration to create sound. Percussion instruments are simplest and can easily be defined as an object produces a sound through being struck. One percussion instrument that is certainly most recognizable is the drum. Drums are usually tuned to a specific note by tightening or loosening the head of the drum (tighter drum higher note). Drums can be arranged by pitch (larger drum lower pitch). String instruments are also some of the simpler that make sounds with vibrating tensioned strings. The strings are vibrated by rubbing a bow against them, striking them, or plucking them. Most stringed instruments have something that amplifies the sound called the sound box or resonator. The sound box is often the largest part of the instrument body with the top part specifically called the sounding board. Different strings and different instruments are able to produce different notes. Pitch is modified by the length, weight (thickness), and tension (tightness) of the string:

Figure 6-15: The Doppler Effect is the apparent change in frequency of a wave due to a change in position.


59

length - longer strings vibrate slower than shorter ones resulting lower notes.

weight - heavy, thick strings make lower notes as compared to lighter, thinner ones.

tension - tight strings makes a higher notes compared to loose ones.

Wind instruments are separated into woodwinds and brass for musical purposes. Each type is basically hollow tubes or pipes of some sort that produce a sound when air is moved through them. Most wind instruments have keys or finger holes to vary the pitch of the sound by shortening or lengthening the air column inside the instrument. For the purposes of this course all instruments of this type will can be divided into two categories: open ended and closed ended resonators (pipes). The term “pipe” refers to any tube regardless of its shape or if additional holes have been cut into it. The air column extends to the first open hole. An open ended resonator has both ends open to the air. A musician would blow in through one end and the sound would come out the other end of the pipe. The air moves through the "pipe" in different ways when different keys are played. A closed ended resonator has one end closed off, and the other end open. The frequencies of sounds made by these resonators are different because of the different ways that air will move at a closed or open end of the pipe. Vibration inside a pipe forms a standing wave which is the result of the sound wave reflecting off the end of the tube (whether closed or open) and interfering with itself. When sound is produced in a resonator, only the waves that will fit in the tube resonate, while other frequencies are lost. The longest wave that can fit in the pipe is the fundamental or first harmonic, while other waves that fit are overtones. Overtones are multiples of the fundamental with areas of highest vibration are called antinodes and areas of least vibration called nodes. In an open pipe, the ends are antinodes. In a closed pipe, the closed end is a node and the open end is an antinode. Thus, closed pipes yield only half the harmonics as compared to an open pipe.

Different amounts of a wavelength in a pipe will result in a different frequency being heard. Although the actual length of the pipe remains the same, different notes are played. The simplest, smallest wave that can fit in a closed end pipe is a quarter of a wavelength. This leads to a formula relating wavelength to length of the pipe:

= 4L

where is wavelength and L is the length of the pipe both measured in meters. This formula is not very helpful so using the wave speed formula and algebraic manipulation another more useful formula is found:

4L

v f

where f is the frequency of sound and v is the velocity of sound in air. Sometimes the above formula is written to allow the harmonic number to be used so that the frequency of a particular harmonic can be found.

4L

vn f

where n is the harmonic. It is important to remember that all of the harmonics in closed end pipes are going to be odd numbers. The fundamental (first harmonic) for an open end pipe needs to be an antinode at both ends, since the air can move at both ends so the smallest wave will be half of a wavelength. This leads to two different formulas:

= 2L

2L

vn f

Open end pipes can have any number harmonic, odd or even.

Physics Honors Light and Optics

60

Chapter 7 – Light and Optics

Visible light is one example of an electromagnetic wave; these waves are sometimes referred to as light even when they cannot be seen with the naked eye. Other commonly used terms are radiant energy and electromagnetic radiation. Light travels in waves that move outward in all directions from its source. But unlike sound, light waves move in straight paths called rays and certain “light” cannot curve around objects. This is why objects block out light rays and cast shadows. However, light can travel through a vacuum, something that sound cannot do.

7.1 Electromagnetic Waves Electromagnetic waves are generated by accelerating or vibrating charges. The electromagnetic wave is a transverse wave consisting of electric and magnetic fields that are perpendicular to one another and are in phase with one another. Electromagnetic waves are self-sustaining since a changing magnetic field produces a changing electric field, and this changing electric field produces a changing magnetic field. Disturbances in electromagnetic fields around particles can travel through space as they move outward, or radiate. This radiation or energy is transferred from the Sun to Earth without the existence of a physical medium between the two bodies. Unlike mechanical waves, electromagnetic waves do not require a medium; they can travel in a vacuum. The speed of an electromagnetic wave in vacuum is referred to as the speed of light and has a value symbolized by the letter c and is equal to 3.00 x 108 m/s.

7.2 The Electromagnetic Spectrum As with other types of waves, the speed of an electromagnetic wave equals the product of its wavelength and frequency. The difference with electromagnetic waves is that the speed is constant if the wave travels in a vacuum. There are a variety of combinations of frequencies and wavelengths that exist; the full collection of arrangements is known as the electromagnetic spectrum. The observed electromagnetic spectrum is divided into several regions (an example can be found in figure 3). Unlike mechanical waves, the energy associated with an electromagnetic wave is related to its frequency. Since spectra are arranged either by increasing or decreasing frequency it is a quick way to compare energies. Radio waves make up the lowest frequency region of practical importance and includes both radio and television waves. Microwaves are commonly used for long-distance communication and cooking. Infrared waves also known as IR are found just below of red light in terms of frequency and include most of the heat given off by common objects. Hand-held remote controls and cell phones often operate using infrared waves. The part of the electromagnetic spectrum that we can see with our eyes is called the visible light region. The pure colors of light—red, orange, yellow, green, blue, and violet—differ from each other in frequency and wavelength. Red light has the longest wavelength and smallest frequency; violet light has the shortest wavelength and greatest frequency. The color that you see when you look at an object is actually the wavelength of light that is reflected by that object. The other visible wavelengths are absorbed by it. Mixtures of light waves of different wavelengths (the pure colors) produce colors other than those listed (including white). Each mixture produces a unique sensation in the eye and a unique color. The primary colors of light – not pigment or paint – are red, green, and blue. Mixing these colors of light can produce all of the colors of the visible spectrum. These three

Figure 7-1: An electromagnetic wave consists of an electric fields (E) and a magnetic field (B) that are at right angles to each other.


61

Figure 7-2: Additive colors of light.

standard additive colors of light can be mixed to produce white light. Two colors that produce white when added together are called complementary colors. The complementary colors are cyan, magenta, and

yellow which are formed by combining the two adjacent primary colors. Red and blue make magenta, blue and green make cyan, and red and green make yellow. The color complement to a primary color is called a secondary color. When the three secondary colors are combined, they produce black.

The frequency range just above visible light is called ultraviolet light or UV. Ultraviolet light from the sun can be harmful over time; most harmful UV rays are blocked by Earth's ozone (O3). X-rays are very penetrating and widely used in medicine to "see" past skin and tissue. The “last” region of the electromagnetic spectrum is for gamma rays. These waves are given off by radioactive materials and are even more penetrating and damaging than x-rays.

7.3 The Dual Nature of Light In the late 1600’s, Danish scientist Christian Huygens considered light as a wave. He envisioned a wave crest advancing by imagining each point along the wave crest to be source point for small, circular, expanding wavelets, which expand with the speed of the wave. Huygens' principle can be used to derive the law of reflection and the law of refraction.

Wavelength (meters)

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100 10

1

1020

1019

1018

1017

1016

1015

1014

1013

1012

1011

1010

109 10

8 10

7

Frequency (hertz)

7.6

9

10

14

6.5

9

10

14

6.1

0

10

14

5.2

0

10

14

5.0

3

10

14

4.8

2

10

14

3.8

4

10

14

Figure 7-3: The electromagnetic spectrum.

Figure 7-4: Huygens’ principle.


62

In the early 1800’s English scientist Thomas Young passed light through two slits cut into an opaque screen. The light produced a pattern of many bright bands, separated by dark hands, on another screen behind the slits. Young explained this phenomenon by postulating that light is actually a wave. By Huygens’ principle, the light from each slit is treated as a source of light spreading out in all forward directions. Therefore, two coherent light waves—waves of the same frequency that are produced by sources in phase - are overlapping coherent waves. This leads to the alternating columns of constructive and destructive interference. A band of light appears where a column of constructive interference meets the screen. Where a column of destructive interference meets the screen a dark spot appears because the two light waves destroy each other. The pattern of bright and dark bands or fringes on the screen reveals the arrangement of the alternating lines of interference; it is referred to as an interference pattern. In a typical setup, such as in figure 6, the distance from the slits to the screen is much larger than the separation between the slits. This leads to the following equations:

bright fringes: d

m θ sin

dark fringes: d

)2

1 - (m θ sin

monochromatic light used: L

xd

where is the angle from either slit to a point on the screen, d is the separation between the slits, m is the number of the fringe, and x is the distance between bright fringes.

When the electric and magnetic net fields resulting from the combination of waves have larger magnitudes than the fields from the individual waves it is called constructive interference; when the combination results in fields of reduced magnitudes we call this destructive interference. Interference effects are noticeable when coherent light – different light waves of the same frequency (monochromatic) that have a constant phase relationship – is used. When monochromatic light, the bands are of equal width and are equally spaced from each other. Waves are in phase when the phase difference corresponds to a whole number multiple of the wavelength. When the phase difference corresponds to an odd number multiple of one- half a wavelength, the waves are said to be completely out of phase. When the phase relationship between the different waves varies randomly, the waves are said to be incoherent.

Figure 7-5: Interference as light passes through an opening proves light in a wave.

Figure 7-6: Young’s experiment.

Figure 7-7: Constructive and destructive interference in Young’s experiment.


63

Figure 7-8: As distance from the source increases, the intensity decreases in an inverse squared relationship.

The term intensity is used to describe the rate at which light spreads over a surface of a given area some distance from a source – not the strength of the source. The intensity varies with the distance from the source and the power of the source. Power is a property of the light source that describes the rate at which light energy is emitted by the source. Light energy emitted by the source travels outward in all directions. As distance from the source increases, the same energy spreads over a wider area.

By the mid-1800s, scientists were convinced that light is an electromagnetic wave with the speed of visible light having the same value as other electromagnetic waves. By the late 1800s, certain experiments showed that light behaved as though it consisted of particles. These particles were thought of as massless bundles with a specific amount of energy and were called photons. These apparently contradictory models of light (wave and particle) took decades to resolve. Today light is explained as having a wave-particle duality which means it is both simultaneously. When electromagnetic radiation strikes certain materials, particularly metals, electrons are ejected from them and escape into the space around the materials. This phenomenon is known as the photoelectric effect. Materials that behave in this manner are said to be photoemissive, and the emitted electrons are referred to as photoelectrons. This effect is the basis of the photocell that powers solar-powered calculators.

The more intense the electromagnetic radiation that strikes a photoemissive material, the more photoelectrons ejected per second. A brighter beam of light causes more photoelectrons to be emitted per second than a dimmer one but increasing the intensity of the radiation does not result in more energetic photoelectrons. The kinetic energy of the emitted electrons depends on the frequency of the incident radiation and on the type of photoemissive material. The higher the frequency, the greater the energy of the photoelectrons.

7.4 Light Incident Upon a Surface When light strikes the surface of an object, three things can happen. Some light may be bounced back, or reflected, by the surface. Some light may be absorbed as heat energy. Some light may be transmitted, passing through the object. A shiny, metal surface reflects much of the light that strikes it whereas light that strikes a black top road is absorbed as heat. Clear glass allows most light to be transmitted through it. We see objects because their surfaces reflect light. A mirror gives an accurate reflection because it has a smooth, shiny surface. When parallel rays of light strike such a surface the reflected rays are also parallel. This type of

1 2 3

Figure 7-10: Light interacts with matter – (1) reflection, (2) absorption, and (3) transmission.

electron light

metal surface

Figure 7-9: Photoelectric effect.


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reflection is called regular or specular reflection. A disturbed puddle of water produces a much different kind of reflection, because its rougher surface spreads the light in a variety of directions. This diffuse reflection allows us to see objects that do not produce their own light but all of the light rays do not remain parallel to each other. Any reflection of light from a surface obeys the law of reflection. This law simply states that the angle that the incident ray makes with the normal to the reflecting surface (angle of

incidence or i) is equal to the angle that the reflected ray makes with the normal to the

surface (angle of reflection or r) and all occur in the same plane. As opposed to reflection where radiation is deflected in one direction, some particles have the ability to scatter radiation in all directions. Scattering of light is usually dependent upon the angle of incidence, the size of particles in the light’s path, and the type or frequency of light.

Two main forms of scattering occur in our atmosphere: (1) Rayleigh scattering and (2) Mie scattering. Rayleigh or selective scattering occurs when certain particles are more effective at scattering a particular frequency of light. Some air molecules are small in size and more effective at scattering higher frequencies (shorter wavelengths) of light. Selective scattering by air molecules is responsible for producing the blue color of our sky on a clear day since blue and violet have the highest frequencies in the visible spectrum. Mie scattering occurs when particles are large enough to scatter all visible light. This is the reason most clouds look white – most of the sunlight that enters the cloud is scattered by the relatively large water molecules. Dark clouds are seen as the cloud becomes thicker and some of the light does not pass through the entire cloud. Objects of different colors absorb light to varying degrees. Dark-colored objects absorb more light as heat energy than do light-colored objects, which reflect more light. For this reason, people usually wear light-colored clothing to keep cool during hot, sunny weather. Materials also differ in their ability to transmit light. Transparent materials, such as window glass, permit almost all of the incoming light to pass directly through them. Translucent materials, such as wax paper, let some light pass through but they scatter the light rays so that images are not transmitted clearly. Opaque materials, like wood and iron, do not allow any light to pass through them.

Figure 7-11: Regular reflection.

surface

i r

Figure 7-13: Law of reflection.

angle of incidence = angle of reflection

Figure 7-14: Rayleigh scattering.

Figure 7-12: Diffuse reflection.


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Generally, an electromagnetic wave’s electric field points in all directions within a plane perpendicular to the wave’s motion. When the electric field is made to vibrate in a specific direction, the waves are said to be polarized. The magnetic field can be polarized as well but traditionally polarization refers to be direction of the electric field. Longitudinal waves, such as sound waves, cannot be polarized since their disturbances are parallel to the direction of wave motion.

7.5 Refraction of Light When light (or any wave) crosses the boundary between two different media at an angle other than 90o, it penetrates into the second medium and travels in a different direction than the incident light; this is known as refraction. The speed of light in the second medium will be different from its original speed in the first medium. The ratio of speed of light in vacuum to its speed in a certain medium is called the index of refraction of the medium. Being the ratio of two speeds, the index of refraction is a dimensionless quantity and is always more than 1 since all speeds will be less than the speed of light.

v

cn

where n is the index of refraction, c is the speed of light in a vacuum, and v is the speed of light in a particular medium. The change in the wave direction at a border or interface between media depends on the difference between the wave velocities in the media. This relationship is expressed in the following mathematical relationship:

2θ sin

1θ sin

2λ1λ

2v

1v

where v1 and v2 represent the speed of the wave in the original medium and the new medium,

respectively, 1 and 2 represent the wavelength

of the original wave and the refracted wave, 1

is the angle of incidence and 2 is the angle of refraction. Since the direction of the light ray changes upon transmission across the boundary, a way to determine the new direction of the ray is required. The direction depends on the change in speed experienced by the light as it goes from the first medium into the second medium. If the speed of light is slower in the second medium, the light bends toward the normal. On the other hand, if the speed of light is greater in the second medium, the light bends away from the normal. The mathematical relationship between

the angle of incidence, 1 (with the index of

refraction n1) and the angle of refraction, 2, (with the index of refraction n2) is called Snell's law (or the law of refraction) and is expressed by:

2θ sin

2n

1θ sin

1n

where n1 is the refractive index for the original

medium, 1 is the angle of incidence, n2 is the

refractive index for the new medium, and 2 is the angle of refraction.

Figure 7-15: (1) Opaque, (2) translucent, and (3) transparent materials.

Figure 7-16: A polarizer is a material that absorbs electromagnetic waves whose electric fields are perpendicular to a certain direction.


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In general, when light is incident on a boundary there is both a reflected and refracted ray. For a given pair of media, when n2 < n1, there is an angle of incidence beyond which no light is transmitted into the second medium; this angle is called the critical angle for total internal

reflection, c. The critical angle is reached when

2 = 90o; therefore 1

2c

n

n θ sin .

Critical angles do not exist when the speed of light is slower in the second medium. The light

then bends toward the normal and away from the boundary. For all angles of incidence greater than the critical angle, no refraction occurs and the light does not enter the second medium. Instead, the rays are reflected back into the first medium, obeying the law of reflection. This phenomenon is known as total internal reflection. The index of refraction depends on the medium and on the frequency of the light that is being refracted. Generally, higher frequencies are refracted through larger angles. This means that light which is made up of a mixture of different colors (polychromatic), such as white light and sunlight, will be separated out into those different colors. If the change between the index of refraction and the frequency is large enough in a particular medium, the separation effect is noticeable such as in water or glass. This phenomenon is known as dispersion and it is responsible for the rainbows that we often see when sunlight passes through water droplets or a glass prism.

Figure 7-17: Refraction.

Incoming light

Normal

Refracted light

1 Original Medium

2 New Medium

Figure 7-18: Measuring angles for Snell’s Law.

C

Glass

Air

Figure 7-19: Critical angle – incoming light ray is in the “more dense” medium.

r

i > C

Glass

Air

Figure 7-20: Total internal reflection – the angle of incidence is greater than the critical angle but the law of reflection still holds true so

i = r.

Figure 7-21: Dispersion.

R O Y G B I V


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7.6 Images In real life a light source typically emits a multitude of rays in different directions. The law of reflection applies to every individual ray of light that strikes a surface. If the surface is smooth, all the reflected rays can be extended to one point behind the surface. To the eye intercepting these reflected rays, the rays all appear to be coming from that point (it looks like the source of light). We say that the eye sees an image of the source behind the surface at that point. When the extensions of the reflected rays behind the mirror intersect what is formed is called a virtual image. The virtual image is upright (erect), left-right reversed, equal in size to that of the object, and each part of the image appears as far behind the surface as the corresponding part of the object is in front of the surface. When the actual reflected light rays themselves intersect (not just their extensions), they form what is called a real image. Real images are always inverted and can be projected on a screen or film in a camera.

7.7 Mirrors There are two general categories of mirrors: plane and spherical. The simplest and most commonly used mirror is a plane mirror. A plane mirror is one that is perfectly flat. When you look at your reflection in a plane mirror what you see is called an image of yourself. The source that is being reflected in the mirror (in this case, you) is called the object. Using the law of reflection, several results about reflection with a plane mirror can be found: The distance between the object and the mirror, called the object distance (do), is equal to the distance between the image, on the opposite (back) side of the mirror, and the mirror, called the image distance (di).

The image is right-side-up, referred to as upright.

The image is the same size as the object.

The image is left-right reversed because the image is facing you. (This is like when a person faces you; their right hand is on your left, and vice versa.)

A spherical mirror has the shape of a section of a sphere. If the reflecting surface is on the outside of this spherical section it is called a convex mirror; if the reflecting surface is on the inside it is called a concave mirror. The principal axis or optical axis of the mirror is the line that passes through the center of the mirror perpendicular to the surface. The center or radius of curvature is the radius of the sphere from which the spherical section of mirror was taken. Light rays that are parallel to the principal axis and are incident on a spherical mirror will either converge at (concave mirror) or diverge from (convex mirror) points a certain distance from the mirror’s surface; this distance is called the focal length (f). The point on the principal axis a distance of one focal length away from surface is called the focal point (F). Some focal points are found in front of the mirror (positive value) and some are behind the mirror (negative value). The main objective in image formation by reflection is to determine the size, location, and orientation of the image of a given object. There are two methods for obtaining this information, ray tracing (geometric) and the mirror equation (algebraic). Ray tracing is used to help visualize the situation and the mirror equation is used to obtain accurate numerical results. The mirror equation is:

f

1

d

1

d

1

io

Figure 7-23: Spherical mirrors use common terms to describe special areas.

Figure 7-22: Plane mirrors create virtual images.


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The magnification of the image is defined to be the ratio of the height of the image (hi) to the height of the object (ho). The result is:

o

i

d

d - M

o

i

h

h M

SUMMARY

f and R are positive for concave mirrors

f and R are negative for convex mirrors

M is positive for upright images

M is negative for inverted images

image height is positive for upright images

image height is negative for inverted images

di is positive for real images (those that are in front of the mirror)

di is negative for virtual images (those that are behind the mirror)

do is positive for real objects

do is negative for virtual objects

7.8 Lenses Since light rays bend upon transmission into a different medium, an image can be achieved by refraction. This way of forming images uses a lens instead of a mirror. A lens is a piece of transparent glass or plastic that has curved surface. The curved surfaces refract light rays that pass through the lens. The shape of a lens determines how it bends light.

Lenses come in many different types. A lens with surfaces that curve outward bends light rays so that they are focused in toward a common point. A lens with surfaces that curve inward bends light rays so that they spread out. These two basic types are a converging lens (convex) that is thicker in the middle than at the edges and a diverging lens (concave) that is thinner in the middle than at the edges. The principal axis of the lens is the line passing through the center of the lens making right angles with the surfaces. Incident rays that are parallel to the principal axis will converge to a focus (for a converging lens) at a focal point F on the axis and at a focal length f from the center of the lens. Images of objects seen through lenses may be larger or smaller than the object itself. For instance, the lens of a camera forms smaller images of objects. A photocopy machine has a lens that forms images the same size as the original object. Binoculars contain lenses that magnify objects, making them appear larger. The relationship between di and do is called the thin-lens equation and is the same as the mirror equation. The magnification of the image also obeys the same relationship that it does for mirrors. SUMMARY

f is positive for converging lenses

f is negative for diverging lenses

M is positive for upright images

M is negative for inverted images

Figure 7-24: Mirror ray tracings.

Figure 7-25: Basic lens types.


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di is positive for real images (those that are on opposite side of the lens from the object)

di is negative for virtual images (those that are on the same side of the lens as the object)

do is positive for real objects (those from which light diverges)

do is negative for virtual objects (those toward which the light converges)

Physics Honors Electricity

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Chapter 8 – Electricity

Electrical energy is the ability to do work by virtue of the forces of the attraction and repulsion between electric charges. Static electricity involves forces exerted on matter because of an imbalance of electric charge. Static electricity is associated with charges at rest. The term “at rest” means that there is no continuous flow of charge in any direction. Current electricity refers to the flow of electric charge through matter. For example, a person shuffling across a carpeted floor (creates a buildup of static charge) and then touching a metal doorknob will often experience a mild shock (allows for a discharge).

8.1 Electric Charge All substances are made up of smaller objects called atoms that contain electric charge. To our knowledge, electric charge is a fundamental property of nature that comes in two types: positive charge and negative charge. Most materials contain an equal number of positive and negative charges and are said to be electrically neutral (having zero net charge). Atoms consist of a small central nucleus that contains positively charged particles called protons. The nucleus is surrounded by an equal number of negatively charged particles called electrons. The SI unit of electron charge is called the coulomb (C), a derived unit. One coulomb is the amount of charge contained in 6.25 x 1018 elementary charges. The magnitude of an elementary charge refers to both protons and electrons. While the sizes of the charge are the same, the signs are opposite; an electron has charge of – e and protons carry a charge of +e. One of the properties of electric charge is that it is quantized. This means that the charge only comes in distinct or discrete units. The smallest available charge is that of a proton or electron. Another property of electric charge is that it is conserved. Therefore, in any physical process electric charge is never created or destroyed; the total electric charge of the universe remains constant.

The positive and negative charges in an object can become separated, usually by the movement of electrons, so that one side of the object contains more of the negative charge and the other side is left with more of the positive charge. Such objects are said to be polarized. In atoms and molecules electrons can be completely removed or extra electrons can be added. An atom with one or more electrons removed will have a net positive charge and is called a cation (positive ion); an atom with extra electrons will have a negative charge and is called an anion (negative ion). It is sometimes necessary to calculate the amount of charge that have been transferred, to do this one would use the following equation

e

qn

where n is the number of elementary charges, q is the excess (or deficient) charge in coulombs, and e is the value of coulombs per elementary charge (1.60 x 10-19 C/e). Important to the practical use of electricity is the fact that in some materials, called conductors, electrons are relatively free to move, such as metals. In other materials, called insulators, electrons are not very free to move, such as rubber, plastic, and wood. There are also materials, called semiconductors, whose behavior is not clearly conducting or insulating. These materials are important in modern technology and can be manipulated to be more conducting or more insulating based on the needs of a situation.

8.2 Detecting Static Charge The presence of excess charge on an object can be detected by bringing the object near an electroscope, a device that consists of a metal knob attached to two light metallic leaves. If either a positively or a negatively charged object is brought near the knob of the electroscope, the electrons within the electroscope are forced to rearrange themselves, and the electroscope becomes electrically polarized. Each leaf acquires the same type of charge as the charged object, and the leaves diverge.


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A neutral electroscope cannot be used to distinguish between a positively charged object and a negatively charged object. However, an electroscope can be given a known charge (either by contact or by induction) and then used to detect both the presence and the type of charge brought near it. The leaves of a charged electroscope are separated due to the charge on the leaves. If an object brought near the knob of the charged electroscope has the same type of charge as the electroscope, the leaves will diverge even more. If the object brought near the charged electroscope is oppositely charged, the leaves will converge to a vertical position. Charging by Induction Electric charges are relatively free to move or rearrange themselves in a conductor which leads to a process called charging by induction. In this process, a charged object is brought near a neutral conductor causing a redistribution of free electrons. For example, if the object in question is negative, the electrons in the conductor will be pushed to the side farther from the charged object. If the conductor is then grounded (connected to an object, like Earth, that can give or receive a large amount of electrons), free electrons can either leave the conductor resulting in a net positive charge once disconnected from the ground. The net charge on the conductor will be of opposite sign to that of the object used to induce the charge.

Charging by Contact Electrons are bound more tightly in some materials than in others. Two dissimilar, neutral objects may become charged when they are rubbed against each other. For example, rubber tends to hold onto electrons more firmly than fur. When a rubber rod is rubbed against a piece of fur, electrons transfer from the fur to the rubber rod. The fur loses electrons and becomes positively charged. The rubber rod gains electrons and acquires a net negative charge. The magnitude of the charge on the fur and the rod is equal and the signs are opposite. If the rubber rod is then brought near a suspended neutral object, the negative charge on the rod repels electrons in the object, forcing them to move to the far side. As a whole, the object remains neutral, but the redistribution of electrons causes it to become electrically polarized. The side of the object closest to the rod becomes positively charged and the side of the object farthest from the rod becomes negatively charged (positive charges do not move, only the electrons move). The rod’s excess electrons attract the protons on the near side of the object and repel the electrons on the far side. The force of attraction is stronger (since the rod is closer to that side of the object) and the object moves toward the rod. When the object touches the rod, some of the rod’s excess electrons transfer to the object. The object gains electrons and becomes negatively charged. This method of charge transfer is called charging by contact. The object and the rod are now both negatively charged and repel each other. The object moves (and remains) as far away from the rod as possible. The net charge on the rod and object will be of the same sign. Figure 8-3: Charging by conduction.

Figure 8-2: “Charging” by induction.

Figure 8-1: Using a charged electroscope.


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8.3 Coulomb's Law Electric charges exert electrostatic forces on each other. The rule for determining this electric force is called Coulomb's law. This law states that for two stationary charges, called q1 and q2, the magnitude of the electrostatic force between them is inversely proportional to the square of the distance between them. An attractive force between the charges is negative and a repulsive force is positive. The formula is as follows:

2r

)2

q 1

k(q

eF

where Fe is electric force between the charged objects, q1 and q2 are the charges of the two objects, r is the distance between the charges objects, and k is the proportionality constant (known as the electrostatic constant) with a

value of 8.99 x 109 2C

2m N .

Two additional rules help to determine the direction of the electric force. One rule is that the force is directed along the line joining the charges. The second rule, called the law of charges, is that like charges repel each other and opposite charges attract each other. There are actually two forces, one exerted on q1 by q2 and one exerted on q2 by q1. These forces have equal magnitudes, found using Coulomb’s law, and point in opposite directions based on Newton's third law of motion. When more than two charges are involved, the force on any one charge can be determined using the principle of superposition. This principle states that the force on any charge is the vector sum of the forces that each of the other charges exert on it individually.

8.4 Electric Fields An electric charge in empty space experiences no electric forces, but an electric charge near another electric charge does. The reason for this difference is that a charge creates an electric field that fills the surrounding space and affects other nearby charges. As long as no other charge is present, an electric field does nothing. When another charge (referred to as a test

charge) is introduced into the field, the field exerts a force of attraction or repulsion on it. The two charges or charged materials apply forces to each other even though they are not in physical contact. The electric field is a vector quantity defined as the force per unit of positive charge (which usually small in magnitude and referred to as a test charge) with the SI unit of N/C. The magnitude of an electric field is referred to as its intensity. The electric field is defined in terms of the force on a positive charge. Therefore, a positive charge experiences a force in the direction of electric field, and a negative charge experiences a force in the opposite direction. To determine the electric field due to more than one charge we can use the principle of superposition.

2r

kqE

where E is the electric field intensity, k is the electrostatic constant, q is the charge creating the electric field, and r is distance between the charge producing the field and a test charge. The following features are common to all electric field diagrams: 1. Field lines are drawn as arrows that always begin on positively charged objects and terminate on negatively charged objects. 2. Field lines never intersect each other. 3. Where field lines meet a charged object, they are perpendicular to the surface of the charged object. 4. The concentration of the field lines at any point indicates the intensity or strength of the field. The intensity of an electric field decreases inversely with the square of the distance from a point charge.

Figure 8-4: Electric field around opposite charges.


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8.5 Electric Current and Potential The flow of electric charge constitutes an electric current. In order to produce a flow of electric charge, you must have a separation of positive and negative charges. The current is determined by the magnitude of charge that flows past a point in one second. The SI unit of current is called an ampere (A) and is equal to 1 C/s. The formula for current is

t

qI

where I is electric current, q is electric charge, and t is time needed for the amount of charge to flow. Currents flow through closed paths called electric circuits. However, if a break occurs in the conducting pathway, it is called an open circuit, and electric charge will not flow. If the flow of current is in one direction around the circuit, the current is called direct current (DC). The direction of the flow of conventional current in a circuit is taken to be the direction in which a positive charge would move. This statement is true even though the actual charge that flows is usually negative – to follow the negative charge in a circuit is called non-conventional current. A complete electrical circuit consists of a source of electrons, a force that pushes the electrons through a load or an appliance (such as a light bulb), a conducting path (such as a wire) through which the electrons can flow back to the source, and a switch for closing or opening the circuit.

Electrons flow from a point of excess electrons to a point of a deficiency of electrons. This condition, referred to as a potential difference, produces an electromotive force (emf, usually

symbolized by ), which can push electrons through a conductor. The electric force is a conservative force. This means that it is useful to define a potential energy associated with this force; we call it the electric potential energy. The change in electric potential energy is defined as the negative of the work done by the electric force. Since work done depends on the force applied,

and the electric force depends on the charge on which it is applied, the change in potential energy must depend on the charge. The electric potential difference is used to discuss the potential energy of electricity without knowing a specific charge. The SI unit of electric potential

difference (V) is the J/C and is called a volt (V). Often, instead of electric potential difference, the quantity V is called the voltage.

q

WV

where V is electric potential or voltage, W is the work done on a charge, and q is electric charge. A commonly used unit of energy based on the formula is the electron-volt (eV). An electron-volt is the energy change experienced by an electron when it accelerates through a potential difference of 1 V, therefore one electron-volt equals 1.60 x 10-19 J. This leads to a direct relationship between voltage and the electric field resulting in the formula:

Ed V

where V and E are defined before and d is distance between the charges.

Two large, flat, conducting plates placed in parallel with opposite charges on them create an electric field. The charges are uniformly distributed along the plates and the plates are close together. Under these conditions, the field can be considered uniform, except near the edges of the plates. A device that stores charge on two conductors separated by an insulator is called a capacitor. Capacitors vary in shape and size, but the basic configuration is two conductors carrying equal but opposite charges. The simplest example of a

Figure 8-5: Electric field on parallel plates.


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capacitor consists of two conducting plates which are parallel to each other and separated by a distance. The magnitude of charge that can be stored is referred to as its capacitance.

V

q C

where C is the capacitance, q is the charge stored, and V is potential difference or electric potential connected to the capacitor. The SI unit of capacitance is a Coulomb/Volt which is called a farad (F). One farad is a large amount of capacitance; therefore, capacitances usually range between picofarads and millifarads.

8.6 Producing Electric Current Some chemical reactions release chemical energy stored within the substances that react. You also know that all forms of energy can be transformed to other forms. Early in the nineteenth century, Alessandro Volta, an Italian physicist, produced the first steady flow of electric current by chemical means. He transformed chemical energy into electrical energy in a device called an electrochemical cell. Today, Volta’s experiments can be repeated by using a device known as a voltaic cell or battery. One type of voltaic cell is the wet cell. A wet voltaic cell is composed of strips of two different metals (called electrodes) in a liquid that conducts electricity (called an electrolyte). As a result of a chemical reaction of one metal with the electrolyte, an excess of electrons accumulates on the metal. This causes the strip of metal to become negatively charged (called an anode). The other electrode, which has a deficiency of electrons, is the positive electrode (called the cathode). When a conductor is connected to the ends of the electrodes (terminals), the excess electrons on the anode flow through the wire to the cathode, which has a deficiency of electrons. The flow of the electrons through the wire constitutes an electric current. In this type of cell, the accumulation of electrons usually continues until any one of the chemicals is used up.

Since a wet cell contains liquid, it is difficult to use as a portable source of electric current. A more convenient and portable source of current is the dry cell. The inside of a dry cell is composed of a chemical paste, which serves as the electrolyte, with a graphite (carbon) rod in the center (the positive electrode). The outer casing of the container is made of metal and serves as the negative electrode. When the electrodes are connected, an electric current flows between them. As in the case of the wet cell, the dry cell produces electric current until any one of the chemicals within the cell is used up.

8.7 Resistance and Ohm's Law Current does not usually flow through a circuit unimpeded. The wires through which the current flows offer some resistance to this flow. As the resistance in a wire increases, a smaller amount of current will flow for a given potential difference across the wire. This relationship is referred to as Ohm's Law. The SI unit of

resistance is called an ohm () and is equal 1 Volt/Ampere.

V = IR where V and I are defined as before and R is the electric resistance. The amount of resistance in a particular piece of wire depends on the size and shape of the wire as well as the type of material out of which the wire is made. Temperature also affects the wire’s resistance. If the temperature of a wire increases, the resistance of the wire increases because of the greater thermal vibrations of the atoms and electrons in the metal. Resistance increases in direct proportion to the length of the wire and decreases inversely proportional to the cross-sectional area of the wire. Resistivity is the measured dependence on the type of material. The resistivity of a material has SI units

of m.

A

LR

where R is electric resistance, is resistivity of the wire, L is the length of the wire, and A is cross-sectional area or thickness of the wire.


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Figure 8-6: Commonly used circuit schematic (diagram) symbols.

Electrical wire

Single cell

Connected wires

Battery (series of cells)

Open switch

Capacitor

Resistor (generalized)

Capacitor (polarized)

Resistor (lamp or bulb)

Ammeter

Voltmeter

Ohmmeter

8.8 Energy and Power The definition of electric potential (potential energy per unit charge) is transferred when an amount of charge moves through a potential difference. Batteries transfer this energy to the charge and resistors squander this energy in the form of heat. The rate at which this energy transfer takes place is the power that is either generated or dissipated by a device in the circuit. The SI unit of electric power is same as the power of any other object, the watt (W).

P = VI where P is electric power, V is electric potential

or voltage, and I is electric current. Since power is work divided by time and work is related to energy, power multiplied by time will give energy. A unit of energy that is commonly used to track the household electricity usage is the kilowatt-hour (kWh). This quantity is equivalent to 3.6 x 106 J.

8.9 Circuits and Measurement Devices A traditional way of representing devices in a circuit is with a circuit diagram or schematic. A switch is a device that allows one to control the flow of charge through a circuit by opening and closing the circuit. A device that uses flowing electric charge to do work is called a load. Light bulbs, televisions, CD players, and the like are all loads. In a, incandescent light bulb, charge is made to flow through a thin wire, called a filament. The moving charges exert forces onto the molecules in the wire that cause them to vibrate more rapidly, and the wire gets so hot that it glows. If too much electricity flows through the wires in your house, they too can heat up and cause a fire. Therefore, household circuits are protected by fuses or circuit breakers that open the circuit if the wires get too hot. A resistor in a circuit represents a circuit element (such as a light bulb or a heater) that contains resistance. Circuit elements can be connected in different ways and the overall resistance (sometimes called equivalent or effective resistance) of the combination depends on how they are connected.

An ammeter is a device specially made for measuring currents in a circuit. An ammeter should be connected in series with the device whose current is sought. Ideally, an ammeter should have zero resistance. In practice, the resistance of an ammeter should be much less than the resistances of other devices in the circuit. A voltmeter is specially designed to measure potential differences across devices in a circuit. A voltmeter should be connected in parallel with the device across which the potential difference is being sought. Ideally, a voltmeter should have infinite resistance. In practice, the resistance of a voltmeter should be much greater than the resistances of other devices in the circuit. An ohmmeter is device that combines a power supply with an ammeter and a voltmeter to measure resistance. An ohmmeter should be connected in parallel with the device across which information is needed. In many practical applications, a device called a multimeter is used to test various circuit components.


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8.10 Kirchhoff's Rules As circuitry becomes more complicated it becomes difficult to analyze the circuit using the simple applications of Ohm's law. Kirchhoff's rules apply the conservation of electric charge and the conservation of energy to circuit analysis. Kirchhoff's first rule uses the idea of conservation of charge. It states that the algebraic sum of all currents meeting at any junction in a circuit must equal zero. It is often referred to as the junction rule. A junction is a point in a circuit where three or more wires meet so that the current in the circuit may take different paths into or out of this point. Using this rule, currents entering the junction are given a positive sign and current leaving the junction are given a negative sign. Application of the rule seems to imply that current direction is known; however, it is not always obvious. In these cases it is sufficient to guess a direction and solve the equations. If the current value of direction guessed turns out to be negative, then the guess was wrong and it actually flows in the opposite direction. Kirchhoff's second rule uses the idea of conservation of energy. The loop rule, as it is often referred, states that the algebraic sum of all potential differences around any closed loop in a circuit must equal zero. When using the loop rule, several things should be remembered: 1. Crossing a battery from the negative terminal

to the positive terminal is a potential increase while crossing it from positive to negative is a potential decrease.

2. Crossing a resistor in the same direction as the current is a potential decrease.

3. Crossing a resistor in the direction opposite to the current is a potential increase.

Applying Kirchhoff’s rules involves several decisions which include:

choosing the directions of the currents in different parts of the circuit

choosing which junctions and loops you will use for your analysis

choosing directions for traversing the loops to generate your loop rule equations.

Utilizing these rules carefully will give a system of equations that can be solved for currents, resistances, and potential differences in circuits.

8.11 Circuit Types A load can be connected to an electric circuit in several ways. In a series circuit, electric charge can travel along only one pathway through the circuit. Some Christmas lights are wired in a series circuit – if one of the lights burns out, the pathway is broken, electric charge stops flowing, and all of the lights go out at once. Resistors in series are connected one after the other (or end to end). Connecting resistors in series has the same effect as making one resistor longer. The equivalent resistance (Req) of a series circuit is defined as the resistance of a single resistor that would produce the same current that is in the circuit. The equivalent resistance of a series circuit is the same as the sum total resistance of all the resistors in the circuit. It is important to remember that the same current flows through resistors that are in series with each other. Therefore, Req is greater than any of the individual resistances that contribute to it.

I3

I2 I1

Figure 8-7: Total current flowing into a junction must equal the current flowing out of the junction

so I1 = I2 + I3.

Figure 8-8: Simplified series circuit.


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SERIES SUMMARY

all components have the same current

total resistance is the sum of individual resistances

total voltage is the sum of individual voltage drops in resistors and should equal that of the source

total power is the sum of the individual power dissipation in resistors

open circuit causes the current to stop in the entire circuit

Resistors in parallel are connected across the same potential difference. In a parallel circuit, the electric charge can flow through the circuit along more than one pathway. Christmas lights wired in a parallel circuit do not all go out at once. If one light burns out, electric charge can flow along one of the other paths that is still unbroken. When the current in a wire enters a junction, a branch or a fork in its path, charge is conserved. For parallel circuits, this means that the total current is the sum of the currents in each path. Energy is conserved, so the sum of the voltage drops around a complete circuit is equal to the emf. This means that the potential difference across each path in a parallel circuit is the same. Connecting resistors in parallel has the same effect as making one resistor wider. The result is that the equivalent resistance of resistors connected in parallel can be found by totaling the sum of the inverses of individual resistances.

Once eqR

1is determined, it is inverted to get the

value of Req in ohms. As a consequence, for resistors in parallel equivalent resistance is smaller than the smallest of the individual resistances that contribute to it. PARALLEL SUMMARY

total current is the sum of individual branch currents

total resistance is the less than individual resistances

all components share the same voltage and should equal that of the source

total power is the sum of the individual power dissipation in resistors

open circuit causes the current to stop only in the effected branch(es)

Circuits are most likely a combination of series and parallel elements. A single set of rules does not apply to each part of the circuit. To resolve the circuit, we first need to identify which parts are series and which parts are parallel. Then we selectively apply series and parallel rules as necessary to determine what is happening. COMBINATION STRATEGY

construct a schematic, if not provided, to help determine which resistors are connected in series versus those connected in parallel

redraw the circuit replacing parallel resistor combinations with a single equivalent resistor – forming a series circuit

determine the total resistance of the new series circuit

find the total current available to the modified circuit using Ohm’s Law

working backwards until you reach the original circuit configuration, determine the individual voltage drops and branch currents

A circuit that employs both a resistor and capacitor is called an RC circuit. The capacitor stores energy for future use and the resistor determined the rate of release. These circuits are important in electronics. Capacitors can be connected in series with other or in parallel. The equivalent capacitance depends upon the type

Figure 8-9: Simplified parallel circuit.

+ -

Figure 8-10: Simplified combination circuit.


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of arrangement. For capacitors connected in parallel, the equivalence is simply the sum of capacitors. When connected in series, the equivalent capacitance diminishes and is calculated in a similar fashion to finding the equivalent resistance of a parallel circuit. RC SUMMARY

total capacitance when connected in series is less than individual capacitors

total capacitance when connected in parallel is the sum of individual capacitors

when charging, current flows through the capacitor so it acts like a wire or a load

once charged, capacitor acts like an open circuit

when initially discharging, current flow from the capacitor so it acts like a source

as time passes during discharge the current decrease and capacitor acts like an open circuit once again

Figure 8-11: Simplified RC circuit.

+

-

Physics Honors Magnetism

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Chapter 9 – Magnetism

Magnetism is a force that acts at a distance and is caused by a magnetic field. This force strongly attracts ferromagnetic materials such as iron, nickel and cobalt. Although individual particles such as electrons can have magnetic fields, larger objects such as a piece of iron can also have a magnetic field, as a sum of the fields of its particles. If a larger object exhibits a sufficiently great magnetic field, it is called a magnet.

9.1 Magnetic Theory Like all other materials, magnets are made up of atoms that have electrons orbiting a nucleus of protons and neutrons. Electrons have a property called spin. This spinning creates a magnetic field with N and S poles, just as the spinning Earth has magnetic poles. In most materials, these tiny fields all point in different random directions, so the bulk material does not have a magnetic field. If most electrons in the shells of an atom spin in the same direction or are aligned, the electrons create miniscule magnetic fields. The atom will then respond to the forces of a magnet and the material is magnetized. If half of the electrons spin one way and the rest spin the other way, they will neutralize each other and the material will not be affected by a magnetic field. The factors that determine the way a material responds to a magnetic field are: (1) the alignment of electrons in the substance (2) the domains within the material. Since the atoms or molecules of a substance need to be aligned, gases and liquids are typically not magnetic, because of the free

motion of the particles. There are some exceptions, especially concerning the plasma state of matter. Typically, all magnets are solid metals. A group of atoms in a metal may become aligned, but various “atom groups” may be misaligned. These “atom groups” are called domains. It is necessary to line up many of the domains in a material in order for it to become a magnet. If a magnet is dropped repeatedly, the magnetic properties may be disturbed.

9.2 Magnetism in Matter The very fact that materials have magnetic properties can be traced back to the fact that the electrons in the atoms of a substance possess little magnetic fields as part of their fundamental character. The structures of some substances are such that there are regions of net magnetic fields. The fields of these regions tend to align when an external magnetic field is applied. Materials, like iron, that can be permanently magnetized, are called ferromagnetic. Ferromagnetic materials can often lose their magnetism at high temperatures. There are two other types of magnetic materials: (1) If a magnet attracts a non-ferromagnetic material, the material is called paramagnetic. The atoms in a paramagnet line up in the direction of an external field. (2) If a magnet repels a non-ferromagnetic material, the material is called diamagnetic. Atoms in a diamagnet line up against an external field. Ferromagnetic materials are strongly attracted by a magnetic force. The elements iron, nickel, cobalt and gadolinium are such materials. The reasons these metals are strongly attracted are

Figure 9-1: The magnetic field of a moving

charged particle.

Figure 9-2: Misaligned domains (top) and aligned

domains (bottom).


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because their individual atoms have a slightly higher degree of magnetism due to their configuration of electrons, their atoms readily line up in the same magnetic direction, and the magnetic domains or groups of atoms line up more readily. Iron is the most common element associated with being attracted to a magnet. Steel is ferromagnetic since it is an alloy of iron and several other metals, giving it greater hardness than iron. Because of its hardness, steel retains magnetism longer than iron. Strongly magnetic ferromagnetic materials like nickel or steel lose all their magnetic properties if they are heated to a high enough temperature. The atoms become too excited by the heat to remain pointing in one direction for long. The temperature at which a metal loses its magnetism is called the Curie temperature or Curie point, and it is different for every metal. Paramagnetic materials are metals that are weakly attracted to magnets. Aluminum and copper are such metals. These materials can become very weak magnets, but their attractive force can only be measured with sensitive instruments. Temperature can affect the magnetic properties of these materials. Paramagnetic materials become more magnetic when they are very cold. The force of a ferromagnetic magnet is about a million times that of a magnet made with a paramagnetic material. Since the attractive force is so small, paramagnetic materials are typically considered nonmagnetic. Certain materials are diamagnetic, which means that when they are exposed to a strong magnetic field, they induce a weak magnetic field in the opposite direction. In other words, they weakly repel a strong magnet. Some have been used in simple levitation demonstrations. Bismuth and graphite are the strongest diamagnetic materials - about eight times stronger than mercury and silver. Other weaker diamagnetic materials include water, diamonds, wood and living tissue. The electrons in a diamagnetic material rearrange their orbits slightly creating small persistent currents, which oppose an external magnetic field.

9.3 Types of Magnets In addition to different types of magnetic materials, there are different types of magnets. There are permanent magnets and temporary magnets. A permanent magnet is one that will hold its magnetic properties over a long period of time. Magnetite is a magnetic material found in nature. It is a permanent magnet, but it is relatively weak. Most permanent magnets are manufactured and are an alloy of iron, nickel and cobalt. Rare-earth permanent magnets are a special type of magnet that can have extreme strength. A temporary magnet is one that will lose its magnetism. A common example of this is a magnetized paper clip attracting other paper clips. MAKING MAGNETS Magnets are made by several methods. (1) Contact: When a bar of magnetic material is stroked in one direction with a magnet, the bar becomes magnetized. According to the theory of magnetism, stroking a magnetic substance properly realigns the domains into a regular north-south arrangement. Some substances such as soft iron do not remain a magnet for very long, while others such as steel remain a magnet for a long period of time. (2) Induction: When a magnetic substance is brought close to a magnet – but does not touch the magnet – it becomes a magnet. According to the theory of magnetism, the presence of a magnet near a magnetic object rearranges the domains in the magnetic object. Most of the north poles point in one direction and most of the south poles point in the opposite direction. (3) Electricity: In 1819, the Danish scientist Hans Christian Oersted discovered that a wire carrying an electric current possesses magnetic properties. In 1820, André Ampère wound a long piece of copper wire, a nonmagnetic substance, into a spring-like coil and attached the ends of the coil to a source of electric current. He found that the coil attracted iron and acted like a bar magnet as long as the circuit was closed. When the circuit was broken, the coil lost its magnetism.


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When a magnetic substance is inserted into a coil of wire and the wire is connected to the poles of a dry cell, the magnetic substance becomes a magnet. Typically DC electricity is used, but AC current will also result in magnetism. If the substance is steel or another hard iron alloy, it becomes a permanent magnet. If the substance is soft iron, it becomes a temporary magnet called an electromagnet whose magnetism will diminish as soon as the current is turned off. This feature makes electromagnets good for picking up and dropping objects. The strength of an electromagnetic field is determined by the amount of current, number of coils of wire, and the distance from the wire. The strength of the magnetic field is proportional to the current in the wire. If you double the current, the magnetic force is doubled. If you wrap the wire into a coil, you increase the magnetic force inside the coil, proportional to the number of turns. In other words, a coil consisting of 10 loops has 10 times the magnetic force as a single wire with the same current flowing through it. The magnetic force decreases with distance. It varies inversely to the square of the distance. For example the force at 2 centimeters from a wire is one-fourth that of the force at 1 centimeter. DEMAGNETIZING MAGNETS Magnets can be demagnetized in several ways all of which disturb the regular arrangement of magnetic domains – a necessary requirement for magnetism. (1) Heat: When a magnet is placed in a flame and heated until it is red hot, it becomes demagnetized.

(2) Contact: When one magnet is stroked by another magnet alternately in one direction and then the other, the stroked magnet becomes demagnetized. Note that this procedure is the opposite of the procedure used in making a magnet by stroking. (3) Hammering or Dropping: When a magnet is repeatedly struck with a hammer or when it is struck against a tabletop or some other hard object, the magnet loses its magnetism.

9.4 Law of Magnetic Poles Iron filings tend to concentrate at the ends of a magnet. It is at these ends, called poles, that the power of a magnet appears to be strongest. All magnets have “positive” and “negative” poles, often called north and south, respectively. Like electric field lines, magnetic field lines go from the positive (north) pole, toward the negative (south) pole. When a bar magnet is suspended horizontally by a string, the magnet usually swings and then comes to rest in an approximate north-south position. The pole pointing toward the north is called the north pole of the magnet, while the pole pointing southward is called the south pole of the magnet When the north pole of a second magnet is brought close to the north pole of the suspended magnet, the two north poles repel each other. If the south pole of a magnet is brought close to the south pole of a suspended magnet, the two south poles also repel each other. On the other hand, when the south pole of a magnet is brought close to the north pole of the suspended magnet, the two poles attract each other.

Figure 9-3: A simple electromagnet.

Figure 9-4: Field lines passing through a bar magnet from north to south.


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These observations are summarized in the law of magnetic poles. This law states that like poles of magnets repel each other and unlike poles of magnets attract each other. Note the similarity between this law and the law of electric charges studied earlier. In both cases, opposite charges or poles attract each other and similar charges or poles repel each other. Experiments indicate that when calculating the attractive or repulsive forces between two magnets two factors are important: (1) the strength of the magnets (2) the distance between the poles. The distance between poles has a more pronounced effect than does the strength of the magnets. For example, if you double the strength of a magnet, you double the attractive or repulsive forces. But, if you double the distance between two unlike poles, the attractive force is decreased to one-fourth of its original strength. If you halve the distance between two unlike poles, the attractive force becomes four times greater than the original strength. This is why a bar magnet becomes stronger when bent in the form of a horseshoe – bending the magnet brings the poles closer together.

9.5 Magnetic Fields A magnet is an object or material that attracts certain metals. It can also attract or repel another magnet. All magnets have North-seeking (N) and South-seeking (S) poles. When magnets are placed near each other, opposite poles attract and like poles repel each other. Magnets come in various shapes - the bar magnet is the most common configuration. Magnets also can be square, spherical, horseshoe, and even shaped like a donut. If an iron plate is put across the N and S poles of a horseshoe magnet it would essentially "short circuit" the effect of the magnetism, such that its strength would not be very great. As soon as the plate was removed, the magnet would regain its full strength. An interesting characteristic of magnets is that when you cut a magnet, each part will have both N and S poles.

The magnetic field is both similar to and different than an electric field. A magnetic field (B) consists of imaginary lines of flux coming from moving or spinning electrically charged particles yet a magnetic field is a dipole field – meaning having two poles. This differs from charges in that a positive or negative electrical charge can be alone. Electrical charges are called monopoles, since they can exist without the opposite charge. The magnetic field of an object can create a magnetic force on other objects with magnetic fields. That force is what we call magnetism. Lines of magnetic flux flow from one end of a magnetic object to the other. By convention, we call one end of the object the N or North-seeking pole and the other the S or South-seeking pole, as related to the Earth's North and South magnetic poles. Magnetic flux is defined as moving from N to S. The unit of magnetic flux is the weber (Wb) which is equal to one

Newtonmeter/ampere. The fact that bar magnets interact with Earth is evidence that Earth has a magnetic field. The poles of Earth's magnetic field are near to Earth's geographic poles. However, the definition of the magnetic poles and how they behave requires that the magnetic pole of Earth that is near to Earth's north pole (called the north magnetic pole) is actually the south pole of Earth's magnetic field.

Figure 9-5: Regardless of size, any magnet has both a north and south pole.

Figure 9-6: Magnetic flux always moves from north to south.


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Magnetic field lines are illustrated using the idea that arrows represent vector quantities. In order to draw magnetic field lines, the direction of the magnetic field must be determined. The rule for the direction of a field is: the direction that the north pole of a compass needle would point if placed at that location. Since a magnetic field exists in three dimensions, it is possible that the field may not be easily drawn on a two-dimensional piece of paper. Two additional

symbols, and , are used when representing magnetic fields. If the field is going into the

page, a series of ’s are used. When lines are

pointing out of the page, a series of dots () are used. You can think about these symbols as arrows appearing from in front or behind: from in

front, you see the tip of the arrow (), and from behind you see the fletching or tail (x). The rules for drawing magnetic field lines are: (1) Magnetic field lines are tangent to the magnetic field at every point. (2) Magnetic field lines point away from the north pole of a magnet and toward its south pole. (3) The number of magnetic field lines is proportional to the magnitude of the field. (4) Magnetic field lines always form closed loops.

9.6 Calculating Magnetic Force The relationship between electricity and magnetism is an important one. A moving particle with an electric charge creates a magnetic field. If that charge is moving through an external magnetic field there will be an attractive or repulsive force. There is a relationship between the movement of the particle through the magnetic field, the strength of that magnetic field and the force on the particle described by the equation:

F = qv(B sin ) In this equation, F is the force on the charge in newtons, q is the electric charge in coulombs, v is the velocity of the charge in meters/second, and B is the magnetic field strength in teslas. A tesla

(T) is defined as one newton/amperemeter. The

sin can be important because it lets us see very quickly that there is no force if a charge moves parallel to a magnetic field and that the greatest

force occurs when a charge moves perpendicular to the magnetic field. If instead of a single moving charge, there was electric current through a wire, the force would result in a magnetic field. This is true since an electric current is just a bunch of moving charges. A situation such as this is described with the formula:

F = IL(B sin ) In this equation, where F and B are defined as before, I is the electrical current in amperes, and L is the length of the wire through the magnetic field in meters. The angle referred to in this equation is the one the wire makes with the magnetic field. To find the direction of the magnetic field around a current carrying wire, use one of Ampere’s “hand rules”. The thumb of your right hand should point in the direction of the current in the wire. The magnetic field would make a circular path around the wire, in the direction that your fingers curl. The strength of the magnetic field is proportional to the strength of the current and is weaker the farther it is from the wire. To determine the direction of the magnetic force, a variation of one of Ampere’s “hand rules” can be used. The fingers of your right hand represent the direction of magnetic field, your thumb points in the direction of the current and your palm points in the direction of the magnetic force.

Figure 9-7: Using a right hand rule to determine the direction of a magnetic field around a current carrying wire.


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A solenoid is an electrical device in which a wire has been wound into a helix. The circular loops are usually tightly packed. Solenoids are sometimes referred to as electromagnets. A solenoid carrying a current produces a nearly uniform magnetic field inside the loops. Ampere’s hand rules can be used to determine either the direction of current flowing through the wire or the north pole of the magnetic core.

9.7 History of Electromagnetic Induction As mentioned earlier, Oersted noticed that when a magnetic compass was brought near a current carrying wire that the needle would line up perpendicular to the wire. Oersted was unable to explain this phenomenon. Ampere continued to experiment with this observation and soon found that not only would a compass needle deflect when brought next to a current carrying wire but also another current carrying wire would have forces exerted on it when brought close to the first. This led Ampere to believe that the current was generating a magnetic field.

In 1831, Faraday found that moving a magnetic through a series of wires “created” a current in the wiring. This is stated as Faraday’s law: a change in magnetic flux induces a current in a loop of conducting material. In 1834, Lenz found that a current flows so that it opposes a change in magnetic flux by creating its own magnetic field. This has become Lenz’s law, which is a special case of the conservation of energy. It is important to note that the current will travel so as to oppose the change in magnetic flux, not to oppose the magnetic flux itself. The electric field in the conductor causes an electromotive force (potential difference) which can be found using the formula:

= EL = BLv

In the compound equation, is used to represent the electromotive force or potential difference in volts, L is the length of the conductor in meters, E is the electric field in newtons/coulomb, and B is the magnetic field in teslas.

9.8 Using Electromagnetic Induction Charges moving in a magnetic field create an electric field, just as charges moving in an electric field create a magnetic field. This is called electromagnetic induction and it is important to humans because it is useful. The two most common applications are the electric generator and the transformer.

Figure 9-10: Oersted’s experiment (left) and Ampere’s law (right).

Figure 9-8: Determining magnetic force using a right hand rule.

Figure 9-9: Using Ampere’s hand rules with a solenoid.


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GENERATORS The electric generator, sometimes called a “dynamo,” is a noisy favorite at outdoor events that need electricity. It uses the principle of electromagnetic induction to convert mechanical energy, usually in the form of a gas-powered motor, into electrical energy. A coil in the generator rotates in a magnetic field. As the magnetic flux through the coil changes, it induces an emf which creates a current. An electric motor has the opposite purpose of an electric generator. Motors are designed to convert electrical energy into mechanical work. Electrical power supplies current to a loop that sits in a magnetic field. This loop then experiences a torque causing the loop to rotate. The rotation of this loop can then be used to do mechanical work such as the turning of a wheel. TRANSFORMERS The transformer converts current of one voltage to current of another voltage; it can increase or decrease the voltage of an AC circuit. A simple transformer consists of two coils wrapped around an iron core. A primary coil, containing NP turns, across which an AC voltage VP is applied. The primary coil is bound by an iron core to a secondary coil, containing Ns turns. By induction there will be an induced potential difference, Vs, across the secondary coil. The relationship between the primary and secondary voltages is given by the transformer equation:

primary in turns of number

secondary in turns of number

primary in emf

secondary in emf

Another helpful equation for transformers is:

primary in turns of number

secondary in turns of number

arysecond in current

aryprim in current

Outside a power plant, a “step-up” transformer, whose primary coil has fewer turns than its secondary coil, increases the voltage of the current that is transported along power lines. Then, before the power enters your house, a “step-down” transformer, whose secondary coil has fewer turns than its primary coil, reduces the voltage. The higher voltage on power lines cutting across the town allows more electricity to be transported quickly to urban centers. The lower voltage within your house renders the electricity safer.

Figure 9-11: A step-up transformer.

Physics Honors Quantum Physics

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Chapter 10 – Quantum Mechanics

The ideas that make up our modern understanding of the atoms have evolved over thousands of years. The end of the nineteenth century signaled the dominance of the modern scientific method in establishing James Clerk Maxwell’s theory of electromagnetic radiation and Newtonian mechanics. American physicist Willard Gibbs formalized vector algebra and vector calculus (his notation is still used today). However, the rapid development of physics also led to many unanswered questions. Attempts to answer these questions lead to the development of modern or quantum physics.

10.1 Atomic Models and Notation The concept of the atom was originally proposed as the indivisible quantity from which things are made. In 1897, J.J. Thomson’s discovery of the electron changed this belief by revealing that atoms must have internal structure. Thomson proposed what is called the "plum-pudding model" of the atom in which negatively charged electrons are embedded in a nearly uniform distribution of positively charged matter. A new picture of atomic structure emerged after Ernest Rutherford scattered positively charged alpha particles from a thin gold foil. Rutherford's model placed all the positive charge and most of its mass, at a small central location, called the nucleus, with the negatively charged electrons moving around the nucleus in orbits. While Rutherford's model seemed more reasonable based on the scattering experiments and yielded a good order of magnitude estimate for the size of the nucleus, it was not consistent with experiments on the light given off by atoms nor was it a stable structure according to Maxwell's electromagnetic theory. By applying a large potential difference across a tube containing a low pressure gas, the atoms of the gas can be made to give off light. Passing this light through a diffraction grating or prism separates the light into different wavelengths producing a line spectrum producing an emission spectrum. However, when light consisting of different wavelengths is passed through a gas, some of the wavelengths from

this light will be absorbed by the gas and the resulting light can then produce a line spectrum that is an absorption spectrum. Each element has its own unique emission/absorption spectrum. Unknown gases can be identified by their emission spectra, much as people can be identified by their fingerprints. Danish physicist Niels Bohr proposed a correction to the Rutherford model by postulating that an atom can only change its energy level in discrete steps and not continuously. An energy level is the quantized amount of energy that an atom may have in each level. Bohr’s model was specific to hydrogen and was a planetary model where the electrons in the shell of an atom are not free to be at any distance from the nucleus, nor are they allowed to have any amount of energy. Instead, they are restricted to certain distances and energy values. Each element has a unique set of allowed orbits, each at a certain distance from the nucleus, and the electrons are confined to those orbits. The total energy of an electron, kinetic plus potential, is determined by the orbit it is in and no other amounts of energy are permitted to the electrons. But there was a problem with the planetary model. According to the classical electromagnetic theory, the centripetal acceleration of the electron should cause the electron to emit radiation until the electron loses all of its kinetic energy. This does not happen; not only is the hydrogen atom stable, but a hydrogen atom consistently has the same radius. Bohr provided clarification by stating that under ordinary conditions the electrons in an atom are in the lowest available orbits or energy levels. However, the orbit and energy of the electrons can be raised through the process of

Figure 10-1: Emission spectrum.


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excitation. This occurs when atoms absorb energy through heating, collision with particles, or irradiation. According to Bohr, an atom will absorb only an amount of energy equal to the difference between allowed energy levels. Other amounts of energy cannot be absorbed since doing so would place the electrons between allowed orbits. Since different atoms have different sets of allowed orbits, they also have different excitation energies. The only exception is the amount of energy equal to or greater than that needed to remove an electron from an atom. This amount of energy known as the ionization potential is equal to the difference between the energy of the electron at infinity and its energy in the ground state. Once the electron is removed, it no longer belongs to the atom, and restrictions on its energy no longer apply.

An electron that jumps to a higher orbit is said to be in an excited state. Soon after becoming excited, such an electron falls to a lower orbit or energy Ievel by emitting and losing some or all of the energy it gained. As in the case of absorbed energy, the amount of energy emitted must be equal to the difference in energy between allowed orbits. The energy lost by a falling electron is emitted in the form of a photon of electromagnetic radiation. When an atom absorbs energy it jumps from the ground state to an excited state. Unlike the ground state, the excited states are not stable. The atom spontaneously drops to a lower energy state and emits a photon in the process. The change in energy levels of the atom is given by the formula:

Ephoton = Ef – Ei

In the equation Ef and Ei are the energies associated with specific energy levels or orbits. These values can be found using an energy level diagram which exists for every element in the periodic table. The energies are given in electronvolts which are units of energy equal to 1.60 x 10-19 J. The most recent model of the atom is based on the principles of quantum mechanics and is referred to as the electron cloud model. The electron cloud model proposes that electrons in atoms do not have precisely described positions

and momenta. Instead, only the probability of finding an electron at a specific position with a specific momentum is provided by the laws of nature. The region of most probable electron location is known as a state and each electron in an atom occupies a state. No more than two electrons can be in the same state at the same time. The electron cloud model does not contradict the Bohr model; it casts it in a different light. For example, the electron cloud model’s most probable position for the single electron in the ground state of a hydrogen atom coincides with Bohr’s lowest allowed orbit. According to the cloud model, the electron is not prohibited from being outside that orbit, but the probability of its being inside the orbit is much greater than its being found outside the orbit. The wavelike nature of electrons has interesting consequences that were investigated by German scientist Werner Heisenberg. To locate a small particle, you must observe the light reflected from it or “touch” it with an instrument. Therefore, measuring the location of a particle necessarily involves a change in its momentum. The more precisely you measure the particle’s location, the less certain you can be about its momentum. Conversely, the more certain you are about the momentum, the less certain you are about the location. This can be summarized by the Heisenberg uncertainty principle, which states that both the position and momentum of a particle cannot be with precision at the same time. The nucleus of a hydrogen atom is a proton. Initially it was thought that the nuclei of all atoms were made up of protons and electrons. In 1932 it was discovered that an electron and a proton can combine to form another elementary

Figure 10-2: Wave of electrons in allowed orbit .


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particle, a neutron, which has neutral charge and a mass slightly greater than that of a proton. The nucleus of an atom is made up of protons and neutrons, also called nucleons. Each element in the periodic table can appear in multiple forms. All atoms with the same number of protons are considered to belong to the same element. However, they may have different numbers of neutrons, so their mass is different. Each of the different forms of the same atom that have different masses but have the same chemical properties is called an isotope. The number of protons in a nucleus is called the atomic number, Z; the number of neutrons is called the neutron number, N. The total number of nucleons is called the mass number, A (A = Z + N). The notation used to specify the composition of the nucleus of a chemical

element X is XA

Z .

Atoms exist in an electrically neutral state – meaning that the number of electrons in each atom is the same as the number of protons. Atoms that have an imbalance of electrons and protons are said to be ions. A negatively charged atom is called an anion and has a surplus of electrons while a positively charged ion is called a cation and has a deficiency of electrons.

10.2 Standard Model of Matter In the 1920s it was believed that there were four fundamental particles: the proton, the neutron, the electron, and the photon. These particles were thought to be unbreakable. However, Niels Bohr observed that energy associated with the process of converting neutrons to protons (beta decay) was not conserved. Wolfgang Pauli, in 1931, and Enrico Fermi a little later proposed that a massless particle was emitted along with the beta (β) particle. Fermi dubbed this particle neutrino, for “little neutron”. The neutrino was experimentally observed in 1956. Other studies found more elementary particles. In 1932, Carl Anderson detected what appeared to be a positively charged electron in cosmic radiation. The particle possessed characteristics similar to the electron, but it had positive charge; it was named the positron or anti-electron.

Discovery of the positron confirmed a mathematical model of elementary particles created by Paul Dirac in 1928. In 1930, Dirac had proposed that most elementary particles came in pairs with similar properties. An example of such a pair was the electron-positron pair. Protons also come in pairs. Because of the similarities, the oppositely charged particles became known as antiparticles. In 1935, Hideki Yukawa proposed a new type of particle, one that carried strong nuclear forces through space in much the way a photon carries electromagnetic forces. The changing model of matter, predictions of undiscovered particles, and the promising future of nuclear energy spurred a flurry of research on subatomic particles. In 1937, Anderson and other physicists discovered another particle, similar to the electron but with greater mass. The new particle was named a muon. In 1947, Cecil Powell discovered a new type of particle, but it was not Yukawa’s particle. These particles were named pions or pi-mesons. Only two months after the pions were announced to the world, physicists were shocked by the discovery of yet another family of particles. The rapid accumulation of many new particles and the ever-evolving mathematical apparatus of theoretical physics resulted in a radical revision of the physical view on the structure of matter. Initially proposed by Murray Gell-Mann and Yuval Ne’eman, became known as the Standard Model. The Standard Model groups all elementary particles into three fundamental families: hadrons, leptons, and force carriers.

Figure 10-3: Mass and energy are conserved when a particle and its antiparticle annihilate each other.


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Force Carriers To the best of our current understanding elementary particles interact through the four fundamental forces of nature. These forces are called the strong nuclear force, the electromagnetic force, the weak nuclear force, and the gravitational force. A nucleus may contain many protons a very small distance apart. This situation leads to large electrostatic repulsion between the protons. From the number of neutrons and protons in a nucleus it is possible to estimate the electromagnetic repulsive force between two protons. These forces are very great because of the proximity between the two like charges. If the electromagnetic force were the primary contributor to the net force on these particles, the nucleus would fly apart. Therefore, there must be an attractive force between the particles inside the nucleus that is sufficiently great to overcome the electrostatic force. The strong nuclear force is a short range, attractive force that binds the nucleus together. Scattering experiments with neutrons and protons showed that the strong force exists only if the nucleons are in contact and that the force is independent of the type of nucleon. The carriers of strong nuclear forces are gluons, which were so named because they hold quarks together. Since neutrons experience the strong nuclear force, but do not experience electrostatic repulsion (being electrically neutral) their presence in a nucleus helps to stabilize the nucleus or hold it together. The most stable nuclei are those with nearly equal numbers of protons and neutrons. The more protons in a nucleus, the less stable it is; no nucleus with more than 83 protons is stable. There is another nuclear force called the weak nuclear force which is effective only at very short distances. As its name implies, the weak nuclear force is much weaker than the strong nuclear force. The force carriers of the weak force are bosons. The W- and Z-bosons are different from other force carriers in that they possess a significant rest mass. W-bosons are extremely short-lived and quickly decay into an electron and an antineutrino. An example of a

particle interaction that involves weak forces is neutron decay also known as beta decay. Other force carriers include photons and gravitons. Photons carry electromagnetic forces and interact with quarks and charged leptons. Gravitons are proposed particles that distribute the gravitational force. It is widely believed that at the beginning of the universe, at the big bang, there was only one fundamental force of nature, called the unified force. As the universe evolved, it is believed that the four forces we detect today separated from each other during processes that can be thought of as cosmological phase transitions. Today, scientists are trying to work backward to develop the theory of this unified force. One successful example is the electroweak theory which has demonstrated that the electromagnetic and the weak nuclear forces are really just different aspects of the same basic force. Leptons Leptons are the smallest of the elementary particles and seem to have no internal structure. Six of the leptons are the electron, muon, tau, electron-neutrino, muon-neutrino, and the tau-neutrino. For each of these leptons there is a corresponding antiparticle making a total of 12 different leptons. There are three heavy leptons - electrons, muons and taus - as well as three kinds of neutrinos that correspond to each of the heavy leptons. With the exception of their neutrinos, they are susceptible to weak, gravitational, and electromagnetic forces only. The rest mass of an election or an anti-electron is 9.11 x 10-31 kg, that is the smallest of the lepton masses with the tau lepton having the greatest mass. The electron, muon and tau leptons have the same charge (-e), which is – 1.60 x 10-19 coulomb. Their antiparticles are oppositely charged at + 1.60 x 10-19 coulomb. None of the neutrinos or anti-neutrinos are electrically charged. Hadrons Unlike most leptons which have weak interactions and no internal structure, hadrons contain subatomic structures called quarks. Quarks are the particles that make up the larger


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particles, such as protons, neutrons and pions. There are six different quark-antiquark pairs. Each matching quark-antiquark pair is called a flavor. The flavors of the quarks are up (u) and down (d), charm (c) and strange (s), and finally, top (t) and bottom (b). These cute names help identify quarks but do not tell anything about other characteristics such as mass or electric charge. Quarks carry electric charges that are fractions of e, either ± 2/3 e or ± 1/3 e depending on the quark. Quarks are not found individually in nature, but only in pairs as mesons, or in triplets as baryons. Hadrons respond to all four fundamental forces. The most common baryons are protons and neutrons. Each of them is composed of a combination of three up and down quarks. In contrast, mesons, such as the pion, are composed of a quark and an antiquark and are much less massive. Like leptons, quarks also come in antiparticle pairs. When two antiparticles collide they annihilate each other and all their energy is converted into gamma radiation. Quarks and leptons combine to form atoms. An atom of hydrogen is the simplest element consisting of one electron (charge of – 1 e) and one proton (+1 e). Like all other leptons, the electron has no internal structure. However, the proton is a baryon that consists of three quarks (u, u, and d). Referring to the reference table, the charges of these three quarks are respectively; +2/3, +

2/3, and – 1/3. Their sum, +1, is the correct value for that of a proton.

10.3 Blackbody Radiation One of the first indications of the need for a theory of quantum physics came from the study of the electromagnetic radiation given off by a blackbody. An ideal blackbody is an object that absorbs all of the electromagnetic radiation that is incident upon its surface. This fact means that a blackbody is a perfect absorber of radiation and, in order to maintain thermal equilibrium, a blackbody is also a perfect radiator of electromagnetic energy. Ludwig Boltzmann investigated the radiation emitted by heated bodies and showed that non-blackbodies always radiate less than blackbodies.

In 1896, Wilhelm Wien discovered a law for blackbody radiation (Wien's displacement law) that described how the energy emitted varies with temperature for short wavelengths. Wien’s formula revealed one of the most useful properties of blackbody radiation - that the distribution of energy given off only depends on the temperature of the object. The fact that Wien’s law worked for blue light, but failed for longer wavelengths, puzzled physicists. To make matters worse, the Rayleigh-Jeans formula described the behavior for long wavelengths but failed for short ones. The dependence of the intensity of blackbody radiation on frequency could not be accurately explained by classical physics. In 1900, German physicist Max Planck proposed a way to explain blackbody radiation. Planck proposed that as the material is heated, the vibrating atoms are not able to change their energy continuously. Instead, they emit packets of energy of definite size, meaning they are quantized. Planck’s hypothesis was initially ignored because the suggestion was unprecedented in classical physics. As more experimental data came in, the hypothesis became accepted and Planck received the Nobel Prize for Physics in 1918. Planck proposed that the energy of a vibrating atom, E, is proportional to its frequency of vibration, f:

E = hf The constant h, later to be called Planck's constant, has a value of 6.63 x 10-34 J•s.

10.4 Photoelectric Effect Planck's idea about electromagnetic radiation being quantized was considered to be a good explanation for blackbody radiation but not a general principle. In 1905, Albert Einstein helped to confirm the value of Planck’s constant and provided support for a quantum theory of energy. He proposed that light consists of little bundles of energy called photons. A photon travels at the speed of light in a vacuum. It has zero mass but does carry energy and momentum. The energy of a photon is proportional to the frequency of the electromagnetic emission, just as Planck’s


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quantum theory predicted. The photon energy also can be written in terms of wavelength because frequency is related to wavelength by the speed of light. The momentum of a photon is inversely proportional to its wavelength and directly proportional to its frequency:

p = c

E=

h=

c

hf

Different types of electromagnetic waves differ in their wavelengths and frequencies. For visible light, each frequency is perceived as a different color. Because the energy of a photon is directly proportional to its frequency, red-light photons, with the lowest frequency, have the lowest energy of visible light photons. Blue and violet light have relatively high frequencies and therefore their photons have high energies. Increasing the intensity of red light doesn’t make a difference because it only increases the number of photons. Even dim blue light will eject electrons because the photons have more energy than red-light photons. In addition to providing a more natural explanation of blackbody radiation, Einstein used his photon hypothesis to explain another phenomenon called the photoelectric effect. This effect occurs when light strikes the surface of a metal and ejects electrons creating a current. The minimum amount of energy needed to remove an electron from the surface of a metal is called the work function (W0) for that metal. It was observed that no electrons are ejected from a metal unless the frequency of the incident light exceeds a certain cutoff or threshold frequency, f0, given by the formula:

f0 = h

W0

It was further found that the maximum kinetic energy of the ejected electrons was independent of the intensity of the incident light and was only a function of its frequency:

KEmax = hf – W0.

10.5 Matter Waves In 1923, French physicist Louis de Broglie proposed that nature has a symmetry when it comes to properties of waves and particles. He claimed that particles of matter also should

exhibit both wave and particle properties. By analogy with the momentum of a photon, de Broglie proposed that the momentum of any particle can be written in terms of Planck’s constant and a wavelength. The wavelength associated with an electron, or any moving particle, is called the de Broglie wavelength. The wavelike properties of electrons can be observed when a crystal is bombarded with a beam of electrons. This led to the formula:

= p

h

where p is the momentum of the object found using its mass and velocity. The Compton effect shows that a collision between a photon and the particle changes the momentum of each. The Compton effect is the result of how the direction and energy of light changes when scattered off of electrons that are initially at rest. Using the photon idea and the conservation of energy and momentum, an accurate explanation of the Compton effect emerged. The Compton wavelength of an electron can be found using the formula:

cm

h

e

10.6 Radioactivity One of the ways in which nuclear forces have been explored is through the study of radioactive isotopes. All nuclei with an atomic number greater than 83, and many isotopes of elements with smaller atomic numbers, are unstable. Large nuclei can be unstable because the electrostatic forces are long-range and the strong nuclear forces are short-range. As the number of protons increases in more massive

Figure 10-4: Compton effect.


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nuclei, the repulsive electrostatic forces increase to a greater extent than the attractive nuclear forces. Thus, more massive nuclei tend to be less stable. An unstable nucleus will disintegrate into one or more different nuclei; when it does it will emit one or more particles. Also, a nucleus in an excited state can make a transition to a lower energy state and emit a high-energy photon. Both of these processes are referred to as nuclear decay and the emission that occurs is called radioactivity. The main types of particles emitted during radioactive decay are:

1. alpha particles () are helium nuclei He4

2 .

2. electrons, called --rays when emitted. 3. positrons, which are anti-electrons having the same mass as an electron but opposite

charge, called +-rays when emitted by nuclei. 4. gamma rays, which are high-energy photons emitted when an excited nucleus decays to a lower energy state. Radioactive decay that emits an alpha particle is called alpha decay. The initial unstable nucleus is called the parent nucleus and the final nucleus is called the daughter nucleus. During alpha decay, a large nucleus loses some of its mass, including two protons. As a result, the element transmutes into another element with a lower atomic number. This process is called transmutation. The daughter nucleus will have two less protons and two less neutrons than the parent nucleus. If the X represents the unstable parent and Y is the daughter, we can write this process as:

He Y X 4

2

4)-(A

2)-(Z

A

Z

Notice that the atomic number and mass number on the left-hand side equals the sum of the corresponding atomic and mass numbers on the right-hand side. Radioactive decay that emits a beta particle

(either + or -) is called beta decay. For the

emission of an electron (-), the basic process is that a neutron decays into a proton and an electron. In addition to electrons, each decomposing neutron also emits another elementary particle, called an antineutrino ( ).

The number of protons in the nucleus increases by one, so beta decay is another example of transmutation. The mass number remains the same, but the atomic number increases:

e Y X -A

1)(Z

A

Z

The process of a positron emission (+) is more complicated, a proton decays to produce a neutron and other particles. In addition to positrons, each decomposing proton emits a neutrino ( ). The positron has the same mass

as an electron but it has the same charge as a proton. The mass number remains the same but the atomic number decreases by one:

e Y X A

1)(Z

A

Z

Radioactive decay that emits a gamma ray

photon () is called gamma decay. This process occurs when a nucleus in an excited state decays to a lower energy state. The emitted particle is a high-energy photon and there is no change in either the number of protons or neutrons in the nucleus. Instead, gamma decay redistributes energy within the nucleus and tends to produce a more stable particle. An excited nucleus is indicated by placing an asterisk by the symbol. Thus we have:

X X A

Z

*A

Z

The rate at which nuclear decay takes place is called the activity. The SI unit of activity is the becquerel (Bq), which is defined as 1 decay/s. A common unit of measure for activity is the curie (Ci) which is defined as 3.7 x 1010 decays/s. The properties of radioactive decay allow it to be used as a method of dating certain objects. This is because, for a given initial number of radioactive nuclei, the fraction of nuclei remaining at a given instant depends on the amount of time that has elapsed. A quantity that characterizes the speed of the decay process of a substance is its half-life. The half-life of a radioactive nucleus is the time interval required for the number of these nuclei to reduce by half. To determine the number of half-lives passed, n, use the following formula:

1/2T

t n

where t is the amount of time elapsed and T½ is the substance’s half-life.


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To find the fraction of original mass of the substance remaining after a particular amount of time, use:

n2

1

10.7 Nuclear Fission and Fusion When heavy radioactive isotopes are bombarded with neutrons, they may split into two smaller atoms in a process is called nuclear fission. The reaction also produces “free” neutrons, in fact there are more neutrons produced than absorbed. This means that fission of one nucleus can contribute to a similar reaction in more nuclei. The continual process of repeated fission reactions caused by the release of neutrons from previous fission is called a chain reaction. The amount of fissionable material needed to create a chain reaction is called the critical mass. The energy released in nuclear fission is very large compared to the energy released in chemical reactions. The ability to control the

chain reaction of U235

92 has led to the modern

use of nuclear power. The kinetic energy of atoms produced in nuclear fission is used as thermal energy to make electric energy in nuclear power plants. Given enough energy, two or more light nuclei combine to form a single nucleus of greater mass, the reaction is called nuclear fusion. As a potential energy source, nuclear fusion is even more powerful than fission. The mass of the newly formed nucleus is less than the sum of the masses of the light nuclei. The difference in mass represents the mass that was converted to energy during the process. Some of this energy provides for greater binding energy per nucleon and, therefore, greater stability of the heavier nucleus formed. An example of a fusion reaction between two isotopes believed to be the source of most stellar energy is:

energy He H H 42

11

31

Nuclei repel each other because each has a positive charge. To force them to interact, they must be given enough kinetic energy to overcome their repulsion. Since the magnitude of repulsion increases with charge, only nuclei

having small positive charges may be used. Ordinary hydrogen has a very low reaction probability. However, the hydrogen isotopes

deuterium ( H21 ) and tritium ( H3

1 ) are useful as

nuclear fuels. Deuterium is obtained from heavy water (deuterium oxide). All water contains some traces of heavy water. These traces can be concentrated to form fuel for fusion.

10.8 Nuclear Binding Energy An interesting fact is that the mass of all stable nuclei, with more than one nucleon, is less than the sum of the masses of the individual nucleons. The difference between the mass of a nucleus and the sum of the masses of its

nucleons is called the mass defect, m. This phenomenon occurs because of the binding energy of the nucleus. The binding energy is a measure of how tightly bound together the nucleons are; its magnitude equals the amount of energy required to break the nucleus apart into its constituent nucleons. This puzzle was solved when Albert Einstein postulated the equivalence of mass and energy in his theory of special relativity. Since energy is removed from nucleons when they come together to form a nucleus, mass is also removed. Although the mass lost is small, it is significant and detectable when compared with the extremely small masses of nuclei. The amount of energy can be calculated using one of the most well-known equations in physics: E = mc2. In this equation, E is energy in joules; m is mass in kilograms; and c is the speed of light. Thus, the mass defect of a nucleus in kilograms times the speed of light squared is equal to the energy lost by the formation of the nucleus in joules. It is customary to express the masses of nuclei and subatomic particles in terms of universal mass units, u. One universal mass unit is equal to 931 MeV (megaelectronvolts).


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Average Heat Constants

Specific Heat

(kJ/kgK)

Melting Point (K)

Boiling Point (K)

Heat of Fusion (kJ/kg)

Heat of Vaporization

(kJ/kg)

Alcohol (ethyl) 2.43 (liq) 156 352 109 855

Aluminum 0.90 (s) 933 2770 396 10500

Ammonia 4.71 (liq) 195 240 332 1370

Copper 0.39 (s) 1356 2840 205 4790

Iron 0.45 (s) 1808 3023 267 6290

Lead 0.13 (s) 601 2013 25 866

Mercury 0.14 (liq) 234 630 11 295

Platinum 0.13 (s) 2045 4100 101 229

Silver 0.24 (s) 1235 2485 105 2370

Tungsten 0.13 (s) 3683 5933 192 4350

ice

Water steam

2.05 (s) 273 --- 334 ---

4.18 (liq) --- 373 --- 2260

2.01 (g) --- --- --- ---

Zinc 0.39 (s) 693 1180 113 1770

Physics Honors Glossary

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A absolute error the difference between an experimental value and the accepted value of a measured quantity absolute zero temperature at which a substance has no kinetic energy per particle absorption spectrum a series of dark lines resulting from the selective absorption of particular frequencies of white light acceleration rate at which velocity changes; this change may be in magnitude, direction or both accepted value the most probable value for a measured quantity accurate description of a measurement that is the same or very close to the accepted value action force one of the pair of forces described in Newton’s 3rd law air resistance friction or drag that acts on something moving through air alpha particle a particle with two protons and two neutrons; equivalent to a helium nucleus alternating current electric current that repeatedly reverses direction, twice each cycle ammeter a device used for measuring electric current when connected in series with an electric circuit ampere SI unit of electric current equal to a flow of one coulomb of charge per second amplitude distance from the midpoint to the maximum of a wave or from the midpoint to the wave’s minimum angle of incidence the angle between an incoming (incident) ray and the normal to the surface at the point where the ray strikes the surface

angle of reflection the angle between a reflected ray and the normal to the surface at the point where the ray bounces from the surface angle of refraction the angle between the ray emerging from the interface of two media and the normal to the interface at the point where the ray emerges anion an ion that has a negative charge; formed when valance electrons are added to an atom anode the electrode where oxidation takes place in an electrochemical cell antimatter material consisting of atoms which are composed of antiparticles such as antiprotons, antineutrons, and positrons antinode position on a standing wave where the largest amplitude occurs antiparticle a particle having identical mass, lifetime, and spin as compared to its associated particle, but with an opposite charge (if charged) and a reversed magnetic moment antiquark the antiparticle of a quark Archimedes’ principle an immersed object is buoyed up by a force equal to the weight of the volume of fluid it displaces atom the smallest particle of an element that is electrically neutral retains all of its properties atomic mass weighted average mass of the isotopes of an element atomic number the number of protons in the nucleus of an atom; unique to an element average speed path distance divided by time interval axis straight line around which an object may rotate or revolve


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B baryon an elementary particle composed of three quarks which can be transformed into a proton or neutron and mesons battery a combination of two or more electrochemical cells; a direct current voltage source which converts various forms of energy into electrical energy beats a throbbing variation in the loudness of sound caused by interference when two tones of lightly different frequencies are sounded together Bernoulli’s principle the pressure in a fluid decreases as the speed of the fluid increases best-fit line a straight or curved line on a graph that approximates the relationship or shows the trend among data points beta particle a high speed electron emitted during radioactive decay blue shift an increase in the measured frequency of light from an approaching source; moving object is coming closer to the observer buoyancy the apparent loss of weight of an object immersed or submerged in a fluid buoyant force the net upward force exerted by a fluid on a submerged or immersed object

C capacitor a device used to store charge in a circuit cathode the electrode where reduction takes place in an electrochemical cell cation an ion with a positive charge; formed when valance electrons were removed from an atom Celsius scale temperature scale with zero as the melt-freeze temperature for water and 100 as the boil-condense temperature for water

centripetal acceleration acceleration directed toward the center of a curved path and produced by uniform circular motion centripetal force a center-directed force that causes an object to move in a curved (sometimes circular) path changing velocity either speed or direction (or both) is changing charge the fundamental electrical property to which the mutual attractions or repulsions between electrons or protons is attributed chemical change a process involving one or more substances altering into new substances chemical property the ability or inability of a substance to combine with or change into one or more new substances circuit any complete path along which charge can flow closed system a group of objects not acted upon by any external forces coefficient of friction a ratio of the force of friction to the normal force, it has no units a high coefficient means object is not likely to move easily a low coefficient is found between slippery surfaces component one of the vectors, often perpendicular, whose sum is a resultant vector compound two or more atoms of the same or different elements bonded to form a larger particle compression (1) a pulse of air (or other matter) that is squeezed together; an area of high density and pressure (2) a decrease in spring length from its equilibrium position condensation a region of maximum compression in a longitudinal wave conduction energy transfer from particle to particle within certain material when the two are in direct contact


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conductivity a property of a material that describes the availability of charges that are free to move under the influence of an electric field conductor material, usually a metal, through which energy (heat or electric charge) can flow conserved term applied to a physical quantity that remains unchanged during interactions conservation of charge the principle that net electric charge is neither created nor destroyed but is transferable from one material to another conservative force name given to a force when work done against it is independent of the path taken constant a variable that is not allowed to change constant proportion the relationship that exists between two quantities when an increase in one causes no change in the other constant velocity the speed of an object does not move faster or slower constructive interference addition of two or more waves when wave crests overlap to produce a resulting wave of increased amplitude; occurs when two waves are in phase convection a means of energy transfer by movement of the heated substance itself, such as by currents in a fluid coulomb SI unit of electric charge equal to an electric current of one ampere passing through a given area in one second Coulomb’s law the relationship among electrical force, charges, and distance: The electrical force between two charges varies directly as the product of the charges and inversely as the square of the distance between them crest one of the places in a transverse wave where the it is highest or the disturbance is greatest critical mass the minimum mass of a sample of fissionable material necessary to sustain a nuclear chain reaction

D data table chart that one enters the independent and dependent variables density a property of a substance equal to its mass per volume deceleration a.k.a.. negative acceleration condition occurring when the final velocity of a moving object is smaller than its initial velocity dependent variable variable that responds changes in to the independent variable derived unit a combination of two or more fundamental units used to simplify notation destructive interference combination of waves where crests of one wave overlap troughs of another, resulting in a wave of decreased amplitude; occurs when the phase difference between two waves is 180

o or one-half wavelength

diffraction the spreading of wave fronts into a region beyond a barrier direct current: electric current whose flow of charge is always in one direction direct squared proportion the relationship that exists between two quantities when an increase in one causes a squared increase in the other directly proportional a relationship that exists between two quantities when an increase in one causes an increase in the other displacement (1) term applied to the fluid that is moved out of the way when an object is placed in fluid (2) the straight line change in an object’s position between its initial and final positions distance the total length of a path traveled by an object Doppler effect: the observed change in frequency and wavelength of a wave due to the relative motion of the source or of the receiver


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drag: a.k.a. air resistance force that opposes forward thrust dual nature of light: light can behave as either particles or a wave dynamics a branch of mechanics dealing with how forces affect an object’s motion

E efficiency the ratio of useful energy output to total energy input; the percentage of the work input that is converted to work output elapsed time time that has passed since the beginning of an event elastic term applied to a material that returns to its original shape after it has been stretched or compressed elastic limit the distance of stretching or compressing beyond which an elastic material will not return to its original shape elastic potential energy energy stored in a spring when work is done in compressing or stretching it elasticity the property of a solid wherein a change in shape is experienced when a deforming force acts on it, with a return to its original shape when the deforming force is removed electric circuit a closed path along which charged particles move electric current the flow of electric charge; measured in amperes electric energy total amount of energy in an electrical circuit; equal to the power consumed and the time of the charge flow electric field a force field that fills that space around every electric charge or group of charges electric field line imaginary line along which a positive test charge would move in an electric field

electric field strength the force per unit charge on a stationary positive test charge in electric field electric power the rate at which electrical energy is converted into another form, such as light, heat, or mechanical energy electrical potential energy a.k.a. voltage energy a charge has due to its location in an electric field; measured in volts electric resistance the opposition a material creates to the flow of electric current through it; measured in ohms electrical force a force that one charge exerts on another. Charges with the same sign repel while charges with opposite signs attract. electrically polarized term applied to an atom or molecule in which the charges are aligned so that one side is slightly more positive or negative than the opposite side electrochemical cell a device uses a redox reaction to produce electrical energy from a chemical reaction electrode a metal conductor used to make a connection with a nonmetallic portion of an electrochemical cell electrolyte an ionic solution that conducts electricity electromagnetic energy energy associated with electric and magnetic fields electromagnetic induction the process of generating a potential difference in a conductor due to the relative motion between the conductor and a magnetic field electromagnetic radiation a form of energy which exhibits wavelike behavior as it travels through space electromagnetic spectrum complete range frequencies and wavelengths of all electromagnetic waves from radio waves to gamma rays electromagnetic wave a wave that consists periodically changing electric and magnetic fields that move at the speed of light in a vacuum


99

electron a fundamental negatively charged subatomic particle of matter of negligible mass that moves about the nucleus electronvolt a unit of energy equal to the work done in moving an elementary charge through a potential difference of one volt electrostatic force a force that one point charge exerts of another electrostatics the study of electric charges at rest element a pure substance made of only one kind of atom that cannot be broken down into simpler substances by physical or chemical means elementary charge the charge equal in magnitude to the charge of one electron or one proton elongation an increase in spring length from its equilibrium position emission spectrum a series of bright lines against a dark background that results from the emission of specific frequencies of radiation energy the property of an object or a system that enables it to do work; measured in joules energy level a stationary state of electrons in an atom which represents a specific amount of energy energy level diagram diagram in which energy levels of a quantized system are indicated by horizontal lines from a zero level entropy a measure of the amount of disorder in a system equilibrium a state of balance in which no net force acts on an object

equivalent resistance a single resistance that could replace several resistors in an electric circuit excitation any process that raises the energy level of electrons in an atom

excited state the condition of an atom when an electron is in any level above the lowest energy level due to the absorption of energy experimental value measurement made during laboratory work which may stand alone or be incorporated into one or more formulas extrapolation the extension of a graphed line beyond the region in which data has been gathered

F Fahrenheit scale the temperature scale with the number 32 assigned to the freezing point of water and the number 212 to the boiling point of water first law of thermodynamics states that heat added to a system is transformed to an equal amount of some other form of energy fluid anything that flows and takes the shape of its container; in particular, any liquid or gas fluid friction the resistance force of a gas or a liquid as an object passes through free-body diagram a diagram showing all the forces acting on an object free fall motion under the influence of gravitational force only force any influence that tends to accelerate an object, i.e. push or a pull forced vibration vibration of object made to vibrate by another vibrating object that is nearby free body diagram a drawing, sketched or to scale, that shows all forces acting concurrently on an object free fall the ideal falling motion of an object acted upon only by gravitational force


100

frequency the number of events (vibrations, oscillations, swings, cycles) per unit time; inverse of period friction the force that acts to resist the motion of an object or materials that are in contact fundamental unit one of a set of units from which all measured quantities can be expressed fulcrum the pivot point of a lever

G gas phase of matter in which the molecules are far away from each other moving freely at higher rates allowing them to collide, & break away from each other generator device that converts mechanical energy into electrical energy by rotating a large coil of wire in a magnetic field graph a visual representation of the data enabling easier interpretation of data gravitational field a region where a test particle would experience a gravitational force gravitational force an attractive force that one object exerts on another due to their masses gravitation potential energy work done on or the energy change of an object due to lifting the object above Earth’s surface gravity the force between the mass of Earth and the mass of any object in its vicinity ground state the condition of an atom when its electron are in the lowest energy level and the atom is neither absorbing nor emitting radiation grounding process allowing charges to move freely along a connection between a conductor and the ground

H hadron a particle that interacts through the four fundamental forces (strong, weak, electromagnetic, gravitational) half-life the time needed for one-half of a radioisotope nuclei to decay into its products heat energy transfer by random molecular motions, resulting in gain or loss of internal energy heat engine a device that changes internal energy to mechanical work Heisenberg uncertainty principle states that it is not possible to know precisely know both the velocity and the position of a particle at the same time hertz SI unit of frequency equal to one vibration per second horizontal component a vector component whose direction is parallel to the horizon or x-axis Hooke’s law the distance of stretch (extension) or squeeze (compression) of an elastic material is directly proportional to the applied force hydraulic device a device that uses liquids to transmit pressure from one point to another

I ideal mechanical system a closed system in which no friction or other nonconservative force is acting impulse a change in an object’s momentum caused by a force incident ray a ray that originates in medium and strikes a boundary or the interface of another medium in parallel term applied to portions of an electric circuit that are connected at two points and provide alternative paths for the current between those two points


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in phase term applied to two or more waves whose crests and troughs arrive at a place at the same time so that their effects reinforce each other in series term applied to portions of an electric circuit that are connected in a row or a line so that the current that goes through one must go through all of them independent variable the variable one changes on purpose during an experiment index of refraction a property of a material medium equal to the ratio of the speed of light in a vacuum to the speed of light in the material medium indirect squared proportion the relationship that exists between two quantities when an increase in one causes a squared decrease in the other induced term applied to electric charge that has been redistributed on an object because of the presence of a charged object nearby induced potential difference the difference in electric potential created in a conductor due to its relative motion in a magnetic field induction the charging of an object without direct contact inelastic term applied to a material that does not return to its original shape after it has been stretched or compressed inertia the tendency of an object to resist changing its state of motion infrasonic sound with pitch too low to be heard by human ear; approximately less than 20 Hz insulator a material that is a poor conductor of energy and that delays the transfer of energy interaction a mutual action between objects where each object exerts an equal and opposite force on the other

interference the overlapping or superposition of waves arriving in a region at the same time interference pattern a pattern formed by the overlapping of two or more waves that arrive in a region at the same time internal energy the total energy, potential and kinetic, possessed in the atoms and molecules within a substance – not of the system as a whole inversely proportional the relationship that exists between two quantities when an increase in one causes a decrease in the other ion an atom with a net electric charge, which is due to the loss or gain of electrons ionization potential the energy needed to remove an electron from an atom to form an ion isotope a form of an element having a particular number of neutrons in the nuclei of its atoms

J joule SI unit of work and energy

K Kelvin scale a temperature scale whose zero is the temperature at which it is impossible to extract any more internal energy from a material kilogram the fundamental SI unit of mass kinetic energy energy of motion kinetic friction the resistance force between two surfaces in motion

L law of conservation of charge states that in a closed, isolated system, the total charge of the system remains the same


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law of conservation of energy states that energy cannot be created or destroyed, only transformed from one form into another, but the total amount of energy never changes law of conservation of momentum states that in the absence of a net external force, the momentum of an object or system of objects is unchanged law of reflection states that the angle of incidence is equal to the angle of reflection law of refraction a.k.a. Snell’s law the mathematical relationship governing the refraction of light as it passes obliquely from one transparent medium to another of different optical density lepton a particle that interacts through the electromagnetic, weak, and gravitational forces lever a simple machine made of a bar that turns about a fixed point or fulcrum lift upward force that overcomes weight of an object linear motion change in position along a straight-line path liquid phase of matter in which the molecules flow freely from position to position sliding past each other taking the shape of its container longitudinal wave a wave in which the vibration is in the same direction as the wave’s motion, rather than at right angles to it

M machine a device for increasing (or decreasing) a force or simply changing the direction of a force magnet a material in which atom’s spinning electrons are aligned with one another magnetic field the region around a magnet or any moving charged object where magnetic force exist

magnetic field strength the number of magnetic flux lines per area passing through a plane perpendicularly to the direction of the lines magnetic flux lines imaginary lines that map a magnetic field magnetic force the force produced by the relative motion of charges magnetism the force of attraction or repulsion between magnetic poles mass the amount of matter; a measure of an object’s inertia mass number a number represents the sum of an atom’s nucleons matter anything that has mass, volume, and shape measurement a comparison between an unknown quantity and a standard mechanical advantage the ratio of output force to input force for a machine mechanical energy energy due to the position or the movement of an object can be potential or kinetic energy (or combination of both) mechanics a branch of physics dealing with forces and their effects in motion medium a substance or matter through which waves travel meson a particle of intermediate mass compose of one quark and one antiquark meter the fundamental SI unit of length momentum a measure of an object’s movement motor a device that converts electrical energy into mechanical energy


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N natural frequency frequency at which an elastic object once energized will vibrate; a minimum amount of energy is needed to continue vibrating at that frequency net force the combination of all the forces that act on an object neutrino a neutral particle with energy and momentum but little mass neutron electrically neutral subatomic particle that is found in the nucleus newton derived SI unit of force Newton's first law a.k.a. law of inertia states that a body continues in its state of rest or of motion in a straight line at a constant speed, unless acted upon by a net force Newton’s second law a.k.a. law of acceleration states that the acceleration produced by a net force on a body is directly proportional to the magnitude of the net force applied Newton's third law a.k.a. law of action-reaction states that whenever one body exerts a force on a second body, the second body exerts an equal and opposite force on the first node any part of a standing wave that remains stationary due to maximum destructive interference nonconservative force any force when work done against it is dependent on the path taken non-ideal mechanical system a system in which a nonconservative force acts normal a line drawn perpendicular to a surface normal force upward contact force that is perpendicular to the surface supporting an object

north magnetic pole area from which magnetic flux is considered to emerge nuclear energy energy released by fission or fusion nuclear fission the splitting of a nucleus into smaller more stable fragments and a large amount of energy nuclear fusion the binding of smaller atomic nuclei into a single larger more stable nucleus nucleon principal building block of the nucleus; either a proton or a neutron nucleus the positively charged center of an atom

O orbital a region around an atom’s nucleus that describes the probable location of an electron ohm SI unit of electric resistance ohm-meter SI unit of resistivity Ohm’s law states that the current in a circuit is directly proportional to the voltage impressed across the circuit and is inversely proportional to the resistance of the circuit opaque material that absorbs light without reemission and do not allow light through out of phase term applied to two waves for which the crest of one wave arrives at a point at the same time that a trough of the second wave arrives oxidation the loss of electrons from atoms

P parabola the path traced by a projectile accelerating only in the vertical direction while moving at a constant horizontal velocity


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parallel two lines that never intersect in the same plane parallel circuit an electric circuit in which devices are connected to the same two points of the circuit, so that any single device completes the circuit independently of the others parallelogram a four-sided figure with opposite sides that are parallel and of equal length pascal SI unit of pressure Pascal’s principle the change in pressure on one part of a confined fluid is equal to the change in pressure on any other part of the confined fluid pendulum a mass or bob attach to one end of a string that is attach at its other end to a pivot percent error the measure of the reliability of an experimental result calculated by dividing the absolute error by the accepted value and multiplying by 100 period the time it takes for one full rotation, revolution, or to-and-fro swing periodic motion a.k.a. simple harmonic motion movement in which acceleration is proportional to the distance from an equilibrium position and directed toward that equilibrium position periodic wave a series if regularly repeated disturbances of a medium perpendicular two lines that intersect to form a 90° angle phase the position of a point on a wave relative to another point on the same wave photoelectric effect a phenomenon in which photoelectrons are emitted from a metal’s surface when light of a specific frequency is incident photon a particle that travels only at speed of light and whose energy is related to the frequency of the radiation in the wave model

pitch how high or low frequency appears to be to an observer physical change a type of change that alters the physical properties of a substance without changing its chemical composition physical property a characteristic of matter be can be observed or measured without changing its chemical composition Planck’s constant the proportionality constant in the relationship between energy and frequency plasma matter that consists of positively charged ions and free electrons polarization the aligning of vibrations in a transverse wave, usually by filtering out waves of other directions positron a particle having the mass of a electron and a positive electric charge potential difference difference in electric potential between two points potential energy energy of position, usually related to the relative position of two objects power rate at which work is done or energy is transformed precise describes several measurements of the same event that are nearly identical pressure force per surface area where the force is normal to the surface principle of flotation states that a floating object displaces a weight of fluid equal to its own weight principle of superposition states that the resultant displacement at any point is the algebraic sum of the individual displacements projectile any object that moves through the air or space, acted on only by gravity (and air resistance, if any)


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proton fundamental positively charged subatomic particle that is found in the nucleus pulley a type of lever with a grooved wheel used to change the direction of a force exerted by a rope pulse a single short disturbance in a medium Pythagorean Theorem the square of the longest side of a right triangle is equal to the sum of the squares of the other sides

Q qualitative data information describing physical characteristics such as odor, shape, or color quantity something that you measure quantitative data numerical information describing physical characteristics such as speed, volume, or weight quantized a condition that restricts a system to the absorption or radiation of energy in fixed amounts quantum a discrete packet of electromagnetic energy quantum theory assumes electromagnetic energy is emitted or absorbed in discrete amounts quark one of the basic particles from which many elementary particles may be formed

R

radiation energy transmitted by electromagnetic waves radioactivity process in which some substances spontaneously emit radiation range (1) the difference between the highest and lowest values in a data set (2) the horizontal distance traveled by a projectile

rarefaction a disturbance in air (or matter) in which the pressure is lowered rate how fast something happens or change per unit of time ray (1) a thin beam of light (2) a straight line drawn at a right angle to a wave front and points in the direction of travel reaction force the force that is equal in strength and opposite in direction to the action force, which acts simultaneously on whatever is exerting the action force red shift a decrease in the measured frequency of radiation from a receding source reduction the gain of electrons by atoms reflected ray a ray that has rebounded from a boundary or interface reflection the rebounding of a pulse or wave as it strikes a barrier refracted ray a ray that results from an incident ray entering a second medium of different optical density refraction the change in direction due to a change in speed at the boundary between two media of different densities relative regarded in relation to something else resistance a measure of opposition to electric current offered by a device or conductor resistivity material characteristic dependent on electronic structure and temperature resistor a device designed to supply a specific amount of resistance resolution the process of resolving a vector into components resonance occurs when the frequency of forced vibration matches an object’s natural frequency


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resultant the vector sum of two or more component vectors revolution motion of an object turning around an axis outside of the object rolling friction the resistance force between a surface and a rolling object rotation the spinning motion that takes place when an object turns about an axis located within the object

S scalar quantity a quantity, such as mass, volume, and time, that can be completely specified by its magnitude, and has no direction schematic diagram diagram that describes an electric circuit using special symbols to represent different devices in the circuit scientific notation a way of expressing quantities that consists of a number equal to or greater than one and less than ten follow by the base ten raised to a power second the fundamental SI unit of time second law of thermodynamics states that heat will never of itself flow from one object to another of higher temperature semiconductor material that can be made to behave as either a conductor or an insulator of electricity series circuit an electric circuit in which devices are arranged so that charge flows through each in turn so that if one part of the circuit should stop the current, it will stop throughout the circuit shadow a shaded region that results when light falls on object and cannot reach the region on the far side of the object significant figures the digits in a measured quantity that are known with certainty plus one digit that has been estimated

slope an inclination of a graphed line determined by the ratio of two ordered pairs solid a phase of matter in which the atoms and molecules vibrate about a fixed position south magnetic pole area where magnetic flux is considered to terminate specific heat the quantity of heat required to raise the temperature of a unit mass of a capacity substance by one degree Celsius spectral line a particular absorbed or emitted frequency that is characteristic of an atom speed how fast something is moving spring constant proportionality constant between the applied force and the movement of a spring standing wave a wave in which parts of the wave remain stationary so that the wave appears not to be traveling; the result of interference between an incident and reflected wave static equilibrium state of an object at rest static friction the resistance force that must be overcome to start an object in motion statics a branch of mechanics that studies forces acting on objects at rest strong force an attractive force between nucleons responsible for a nucleus’s stability superconductor material that has infinite conductivity at very low temperatures, so that charge flows through it without resistance superposition occurs when two or more waves travel simultaneously through a medium surface tension force on the particles at the surface of a liquid that causes the liquid to form round droplets


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switch a device that changes the connections in an electric circuit

T tangent line perpendicular to the radius of a circle temperature the property of a material that tells the average energy of particles in the material thermal energy a.k.a. heat total kinetic energy possessed by the individual particles in an object thermal equilibrium the state of two or more objects or substances in contact when they have reached a common temperature thermodynamics study of heat and its transformation into mechanical energy translucent material that scatters light that is transmitted through the material causing blurry images transmutation the conversion of an atom of one element into an atom of another transparent material that allows light to pass through in it in straight lines transverse wave a wave with vibration at right angles to the direction the wave is traveling trough one of the places in a wave where the wave is lowest

U ultrasonic sounds above the normal upper limit of human hearing sounds with frequency; approximately sounds over 20,000 Hz unbalanced force a non-zero net force acting on an object

uniform circular motion the motion of an object travelling in a circular path at constant speed uniform motion the motion of an object travelling at constant speed universal mass unit a.k.a. atomic mass unit one-twelfth the mass of a carbon-12 atom

V vacuum a region of empty space vector an arrow whose length represents the magnitude of a quantity and whose direction represents the direction of the quantity vector components two or more concurrent vectors whose sum is the resultant vector vector quantity a quantity, such as velocity or force, that has both magnitude and direction velocity rate of motion together with the direction of motion vertical component a component vector is parallel to the y-axis vibration an oscillation, or repeating back-and-forth motion, about an equilibrium position volt SI unit of electric potential difference voltaic cell type of electrochemical cell that converts chemical energy into electrical energy voltmeter a device that measures potential difference across an element when connected in parallel with the element in an electric circuit voltage source a device, such as a dry cell or generator, that provides a potential difference volume the amount of space matter occupies


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viscosity a fluid’s resistance to flow

W watt SI unit of power

wave a disturbance that repeats regularly in space and time and that is transmitted from one place to the next with no actual transport of matter wave front all adjacent points on a wave that are in phase wavelength the distance from the top of the crest of a wave to the top of the following crest; the distance between successive identical parts of the wave weight a gravitational force with which a planet attracts a mass work the transfer of energy to an object when the object moves in the direction of an applied force work-energy theorem states that whenever work is done, energy changes

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