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PHYSICS Laboratory I
LABORATORY MANUAL
Name:
Department:
2017-2018
2
Contents
İçindekiler
Contents ................................................................................................................................................. 2
Report Format ......................................................................................................................................... 5
UNIT SYSTEMS ......................................................................................................................................... 7
Measurements and Uncertainty ........................................................................................................... 10
The Vernier Caliper ........................................................................................................................... 12
Experiment 1: Hooke’s Law—Measuring Forces ................................................................................... 15
Experiment 2: Adding Forces Resultants and Equilibrants ................................................................... 18
Experiment 3: Resolving Forces—Components .................................................................................... 22
Experiment 4: Torque—Parallel Forces ................................................................................................. 27
Experiment 5: Center of Mass ............................................................................................................... 32
Experiment 6: Sliding Friction................................................................................................................ 35
Experiment 7: Simple Harmonic Motion: Mass on a Spring ................................................................. 39
Experiment 8: Simple Harmonic Motion—the Pendulum .................................................................... 43
Experiment 9: Conservation of Angular Momentum Using a Point Mass ............................................ 46
Experiment 10: Rotational Inertia of Disk and Ring .............................................................................. 48
Deney 11 : Eğik Atış Rotası .................................................................................................................... 55
Experiment 12: Conservation of Angular Momentum .......................................................................... 59
Experiment 13: Projectile Motion ......................................................................................................... 64
Experiment 14: Projectile Motion Using Photogates ............................................................................ 69
Experiment 15: Polarization of Light ..................................................................................................... 72
Experiment 23. Interference and Diffraction of Light ........................................................................... 81
Experiment 24: Radioactivity Simulation / Rolling Dice Experiment .................................................... 90
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GENERAL INSTRUCTIONS
1. You must arrive on time since instructions are given and announcements are made at the start
of class.
2. You will do experiments in a group but you are expected to bear your share of responsibility
in doing the experiments. You must actively participate in obtaining the data and not merely
watch your partners do it for you.
3. The assigned work station must be kept neat and clean at all times. Coats/jackets must be
hung at the appropriate place, and all personal possessions other than those needed for the lab
should be kept in the table drawers or under the table.
4. The data must be recorded neatly with a sharp pencil and presented in a logical way. You
may want to record the data values, with units, in columns and identify the quantity that is being
measured at the top of each column.
5. If a mistake is made in recording a datum item, cancel the wrong value by drawing a fine line
through it and record the correct value legibly.
6. Get your data sheet, with your name, ID number and date printed on the right corner, signed
by the instructor before you leave the laboratory. This will be the only valid proof that you
actually did the experiment.
7. Each student, even though working in a group, will have his or her own data sheet and submit
his or her own written report, typed, for grading to the instructor by the next scheduled lab
session. No late reports will be accepted.
8. Actual data must be used in preparing the report. Use of fabricated, altered, and other
students’ data in your report will be considered as cheating.
9. Be honest and report your results truthfully. If there is an unreasonable discrepancy from the
expected results, give the best possible explanation.
10. If you must be absent, let your instructor know as soon as possible. Amissed lab can be
made up only if a written valid excuse is brought to the attention of your instructor within a
week of the missed lab.
11. You should bring your calculator, a straight-edge scale and other accessories to class. It
might be advantageous to do some quick calculations on your data to make sure that there are
no gross errors.
12. Eating, drinking, and smoking in the laboratory are not permitted.
13. Refrain from making undue noise and disturbance.
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Semester I Student’s
NAME - SURNAME :
NO :
DEPARTMENT :
DATE: EXP NAME.: INSTRUCTOR’S SIGNATURE
1st Week
2nd
Week
3rd
Week
4th Week
5th Week
6th Week
7th Week
8th Week
9th Week
10th Week
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Report Format
The laboratory report must include the following:
1. Title Page: This page should show only the student’s name, ID number, the name of the
experiment, and the names of the student’s partners.
2. Objective: This is a statement giving the purpose of the experiment.
3. Theory: You should summarize the equations used in the calculations to arrive at the results
for each part of the experiment.
4. Apparatus: List the equipment used to do the experiment.
5. Procedure: Describe how the experiment was carried out.
6. Calculations and Results: Provide one sample calculation to show the use of the equations.
Present your results in tabular form that is understandable and can be easily followed by the
grader. Use graphs and diagrams, whenever they are required.
It may also include the comparison of the computed results with the accepted values together
with the pertinent percentage errors. Give a brief discussion for the origin of the errors.
7. Conclusions: Relate the results of your experiment to the stated objective.
8. Data Sheet: Attach the data sheet for the experiment that has been signed by your instructor.
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Introduction
The aim of the laboratory exercise is to give the student an insight into the significance of the
physical ideas through actual manipulation of apparatus, and to bring him or her into contact
with the methods and instruments of physical investigation. Each exercise is designed to teach
or reinforce an important law of physics which, in most cases, has already been introduced in
the lecture and textbook. Thus the student is expected to be acquainted with the basic ideas and
terminology of an experiment before coming to the laboratory.
The exercises in general involve measurements, graphical representation of the data, and
calculation of a final result. The student should bear in mind that equipment can malfunction
and final results may differ from expected values by what may seem to be large amounts. This
does not mean that the exercise is a failure. The success of an experiment lies rather in the
degree to which a student
has:
• mastered the physical principles involved,
• understood the theory and operation of the instruments used, and
• realized the significance of the final conclusions.
The student should know well in advance which exercise is to be done during a specific
laboratory period. The laboratory instructions and the relevant section of the text should be read
before coming to the laboratory. All of the apparatus at a laboratory place is entrusted to the
care of the student working at that place, and he or she is responsible for it. At the beginning of
each laboratory period it is the duty of the student to check over the apparatus and be sure that
all of the items listed in the instructions are present and in good condition. Any deficiencies
should be reported to the instructor immediately.
The procedure in each of these exercises has been planned so that it is possible for the prepared
student to perform the experiment in the scheduled laboratory period. Data sheets should be
initialed by your instructor or TA. Each student is required to submit a written report which
presents the student’s own data, results and the discussion requested in the instructions.
Questions that appear in the instructions should be thought about and answered at the
corresponding position in the report. Answers should be written as complete sentences.
If possible, reports should be handed in at the end of the laboratory period. However, if this is
not possible, they must be submitted no later than the beginning of the next exercise OR the
deadline set by your instructor.
Reports will be graded, and when possible, discussed with the student.
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UNIT SYSTEMS
SI UNITS
Sciences are built upon measurements. Measurements are expressed with numbers. This allows
the logic, precision and power of mathematics to be brought to bear on our study of nature.
Units of measurement are names which characterize the kind of measurement and the standard
of comparison to which each is related. So, when we see a measurement expressed as "7.5 feet"
we immediately recognize it as a measurement of length, expressed in the unit "foot" (rather
than other possible length units such as yard, mile, meter, etc.) Since many possible units are
available for any measurement it is essential that every measurement include the unit name. A
statement such as "the length is 7.5" is ambiguous, and therefore meaningless.
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THE METRIC SYSTEM
As early as 1670, European scientists were recommending reform of the chaotic unit systems
then in use: systems which differed from country to country. The reformers urged (l) uniformity
and universality, (2) simple ratios of sizes of units, (3) rational relations between units, and (4)
units referenced to constants of nature (such as the circumference of the earth, boiling point of
water, etc.)
In 1791, in the aftermath of the Revolution, the French National Assembly adopted a more
rational system based upon decimal ratios. This came to be known as the metric system. In the
United States, at this time, there was also interest in reform of units and standards. In 1786
Congress approved a decimal system of coinage.
In 1790 Congress considered a report on units which Secretary of State Thomas Jefferson had
prepared at the urging of George Washington. In the report Jefferson proposed, as one
alternative, a decimal system of weights and measures. His system had several unfortunate
features, (1) it retained some of the old unit names (pound, foot, inch, furlong, mile, etc.) but
assigned them new sizes (1 foot contained 10 inches, for example), and (2) his system was not
fully compatible with the metric system then being developed in France. Congress, confused
and ill-informed (as usual) took no action on the proposal.
John Quincy Adams' 1821 Report Upon Weights and Measures was an exhaustive study,
presenting pros and cons of unit reform. Though praising the virtues of the French Metric
system (and noting some shortcomings) he concluded that the U. S. had not attained sufficient
maturity to require adoption of the system. Further he noted that the states had laws of weights
and measures which were substantially uniform. To impose a new system on all states would
raise sticky questions of states' rights.
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So our best opportunity for adoption of a sensible unit system slipped by. While other countries,
one by one, adopted the metric system, the U. S. arrogantly went its own way, feeling no need
nor desire to adopt a "foreign" system of units.
In 1890, metric units were established as the legal basis of all weights and measures in the
United States, but this did little to establish the use of the metric units in industry, commerce,
and everyday life. Today the revised and standardized metric system, called the international
system (SI, for System‚ International) is used in nearly all countries. The United States, South
Africa and less than a dozen non-industrialized countries have not made a commitment to
convert fully to the metric system.
But change is coming"slowly. Several states have marked their highway distance signs in both
miles and kilometers. Some radio stations report daily temperatures in both degrees Fahrenheit
and Celsius. U. S. automakers use metric parts in auto engines. Science and medicine have been
almost exclusively metric in all countries for many years. Other industries use metric standards
because of international trade and competition.
The relative simplicity of the metric system is well illustrated by comparing measurements of
length in the United States System with those of the metric system.
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Measurements and Uncertainty
A measurement result is complete only when accompanied by a quantitative statement of its
uncertainty. The uncertainty is required in order to decide if the result is adequate for its
intended purpose and to ascertain if it is consistent with other similar results." National Institute
of Standards and Technology
No measuring device can be read to an unlimited number of digits. In addition when we repeat
a measurement we often obtain a different value because of changes in conditions that we
cannot control. We are therefore uncertain as to the exact values of measurements. These
uncertainties make quantities calculated from such measurements uncertain as well.
Finally we will be trying to compare our calculated values with a value from the text in order to
verify that the physical principles we are studying are correct. Such comparisons come down to
the question "Is the difference between our value and that in the text consistent with the
uncertainty in our measurements?".
The topic of measurement involves many ideas. We shall introduce some of them by means of
definitions of the corresponding terms and examples.
Sensitivity - The smallest difference that can be read or estimated on a measuring instrument.
Generally a fraction of the smallest division appearing on a scale. About 0.5 mm on our rulers.
This results in readings being uncertain by at least this much.
Variability - Differences in the value of a measured quantity between repeated measurements.
Generally due to uncontrollable changes in conditions such as temperature or initial conditions.
Range - The difference between largest and smallest repeated measurements. Range is a rough
measure of variability provided the number of repetitions is large enough. Six repetitions are
reasonable. Since range increases with repetitions, we must note the number used.
Uncertainty - How far from the correct value our result might be. Probability theory is needed to
make this definition precise, so we use a simplified approach.
We will take the larger of range and sensitivity as our measure of uncertainty.
Example: In measuring the width of a piece of paper torn from a book, we might use a cm ruler
with a sensitivity of 0.5 mm (0.05 cm), but find upon 6 repetitions that our measurements range
from 15.5 cm to 15.9 cm. Our uncertainty would therefore be 0.4 cm.
Precision - How tightly repeated measurements cluster around their average value. The
uncertainty described above is really a measure of our precision.
Accuracy - How far the average value might be from the "true" value. A precise value might not
be accurate. For example: a stopped clock gives a precise reading, but is rarely accurate.
Factors that affect accuracy include how well our instruments are calibrated (the correctness of
the marked values) and how well the constants in our calculations are known. Accuracy is
affected by systematic errors, that is, mistakes that are repeated with each measurement.
Example: Measuring from the end of a ruler where the zero position is 1 mm in from the end.
Blunders - These are actual mistakes, such as reading an instrument pointer on the wrong scale.
They often show up when measurements are repeated and differences are larger than the known
uncertainty. For example: recording an 8 for a 3, or reading the wrong scale on a meter..
Comparison - In order to confirm the physical principles we are learning, we calculate the value
of a constant whose value appears in our text. Since our calculated result has an uncertainty, we
will also calculate a Uncertainty Ratio, UR, which is defined as
UR = |experimental value − text value|
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Uncertainty
A value less than 1 indicates very good agreement, while values greater than 3 indicate
disagreement. Intermediate values need more examination. The uncertainty is not a limit, but a
measure of when the measured value begins to be less likely. There is always some chance that
the many effects that cause the variability will all affect the measurement in the same way.
Example: Do the values 900 and 980 agree?
If the uncertainty is 100 , then U R = 80/100 = 0.8 and they agree,
but if the uncertainty is 20 then U R = 80/20 = 4 and they do not agree.
Graphical Representation of Data
Graphs are an important technique for presenting scientific data. Graphs can be used to suggest
physical relationships, compare relationships with data, and determine parameters such as the
slope of a straight line.
There is a specific sequence of steps to follow in preparing a graph. (See Figure 1 )
1. Arrange the data to be plotted in a table.
2. Decide which quantity is to be plotted on the x-axis (the abscissa), usually the independent
variable, and which on the y-axis (the ordinate), usually the dependent variable.
3. Decide whether or not the origin is to appear on the graph. Some uses of graphs require the
origin to appear, even though it is not actually part of the data, for example, if an intercept is to
be determined.
4. Choose a scale for each axis, that is, how many units on each axis represent a convenient
number of the units of the variable represented on that axis. (Example: 5 divisions = 25 cm)
Scales should be chosen so that the data span almost all of the graph paper, and also make it
easy to locate arbitrary quantities on the graph. (Example: 5 divisions = 23 cm is a poor choice.)
Label the major divisions on each axis.
5. Write a label in the margin next to each axis which indicates the quantity being represented
and its units.Write a label in the margin at the top of the graph that indicates the nature of the
graph, and the date the data were collected. (Example: "Air track: Acceleration vs. Number of
blocks,
12/13/05")
6. Plot each point. The recommended style is a dot surrounded by a small circle. A small cross
or plus sign may also be used.
7. Draw a smooth curve that comes reasonably close to all of the points. Whenever possible we
plot the data or simple functions of the data so that a straight line is expected. A transparent
ruler or the edge of a clear plastic sheet can be used to "eyeball" a reasonable fitting straight
line, with equal numbers of points on each side of the line. Draw a single line all the way across
the page. Do not simply connect the dots.
8. If the slope of the line is to be determined, choose two points on the line whose values are
easily read and that span almost the full width of the graph. These points should not be original
data points. Remember that the slope has units that are the ratio of the units on the two axes.
9. The uncertainty of the slope may be estimated as the larger uncertainty of the two end
points,
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𝑚 = 𝑠𝑙𝑜𝑝𝑒 =(𝑦2 − 𝑦1)
(𝑥2− 𝑥1)
The Vernier Caliper
A vernier is a device that extends the sensitivity of a scale. It consists of a parallel scale
whose divisions are less than that of the main scale by a small fraction, typically 1/10 of a
division. Each vernier division is then 9/10 of the divisions on the main scale. The lower
scale in Fig. 2 is the vernier scale, the upper one, extending to 120 mm is the main scale.
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The left edge of the vernier is called the index, or pointer. The position of the index is what is
to be read. When the index is beyond a line on the main scale by 1/10 then the first vernier line
after the index will line up with the next main scale line. If the index is beyond by 2/10 then the
second vernier line will line up with the second main scale line, and so forth.
If you line up the index with the zero position on the main scale you will see that the ten
divisions on the vernier span only nine divisions on the main scale. (It is always a good idea to
check that the vernier index lines up with zero when the caliper is completely closed. Otherwise
this zero reading might have to be subtracted from all measurements.)
Note how the vernier lines on either side of the matching line are inside those of the main scale.
This pattern can help you locate the matching line.
The sensitivity of the vernier caliper is then 1/10 that of the main scale. Keep in mind that the
variability of the object being measured may be much larger than this. Also be aware that too
much pressure on the caliper slide may distort the object being measured.
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Experiment 1: Hooke’s Law—Measuring Forces
EQUIPMENT NEEDED:
– Experiment Board
– Spring Scale
– Mass Hanger (1)
– Masses
Introduction
The concept of force is defined in Newton’s second law as F = ma; Force = Mass x
Acceleration. Using this law, a force can be determined by measuring the acceleration it
produces on a body of known mass. However, this method is rarely practical. A more
convenient method is to compare the unknown force with an adjustable force of known
magnitude. When both forces are applied to an object, and the object is not accelerated, the
unknown force must be exactly opposite in both magnitude and direction to the known force.
With this statics system, there are two methods of measuring and applying forces. One method
is to hang the calibrated masses. For a mass m, gravity pulls it downward with a magnitude
F = mg, where g is the acceleration caused by gravity (g= 9.8 m/s2 downward, toward the
center of the Earth). The Spring Balance provides a second method of applying and measuring
forces. In this experiment you will use the known forces provided by the calibrated masses to
investigate the properties of the Spring Balance.
Setup
Hang the Spring Balance on the Experiment Board. Be sure the spring hangs vertically in the
plastic tube. With no weight on the Spring Balance, adjust the zeroing screw on the top of the
Spring Balance until the indicator is aligned with the 0 m mark on the centimeter scale of the
Balance as shown in Figure 1.1a
Procedure
Hang a Mass Hanger with a (.02 kg) Mass from the Spring Balance. Measure the spring
displacement on the scale as shown in Figure 1.1b. Record this value in the appropriate space in
Table 1.1. Be sure to include the mass of the Mass Hanger (.005 kg) in the total mass.
By hanging additional masses from the Mass Hanger, adjust the total mass hanging from the
Spring Balance to each of the values shown in the table. For each value, record the spring
displacement.
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Using the formula F = mg, determine the total weight in newtons for each set of masses that was
used. Record your results in the table. (To get the correct force in newtons, you must use the
mass values in kilograms.)
NOTE: When using hanging weights to measure force, a unit for mass is often used as if it were
a unit of weight. Remember, there is a difference between weight and mass. That is: Weight =
Mass x (the Acceleration due to gravity).
Weight is a force that depends on mass and gravity. If the gravitational constant changes—on
the moon, for example—the weight changes as well, but the mass remains the same.
Calculations
À On a separate sheet of paper, construct a graph of Weight versus Spring Displacement with
Spring Displacement on the x-axis (see Figure 1.2). Draw the line that best fits your data points.
The slope of the graph is the spring constant for the spring used in the Spring Balance.
Measure the spring constant from your graph. Be sure to include the units (newtons/meter).
Spring Constant = _____________ (newtons/meter)
Questions
The linear relationship between force and displacement in springs is called Hooke’s Law. If
Hooke’s Law were not valid, could a spring still be used successfully to measure forces? If so,
how?
In what way is Hooke’s Law a useful property when calibrating a spring for measuring forces?
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Experiment 2: Adding Forces Resultants and Equilibrants
EQUIPMENT NEEDED:
– Experiment Board – Spring Balance
– Degree Scale – Force Ring
– Pulleys (3) – Mass Hangers (3)
– Masses – String
Theory
In Figure 2.1, spaceships x and y are pulling on an asteroid with forces indicated by vectors Fx
and Fy. Since these forces are acting on the same point of the asteroid, they are called
concurrent forces. As with any vector quantity, each force is defined both by its direction, the
direction of the arrow, and by its magnitude, which is proportional to the length of the arrow.
(The magnitude of the force is independent of the length of the tow rope.)
The total force on the asteroid can be determined by adding vectors Fx and Fy. In the
illustration, theparallelogram method is used. The diagonal of the parallelogram defined by Fx
and Fy is Fr, the vector indicating the magnitude and direction of the total force acting on the
asteroid. Fr is called the resultant of Fx and Fy.
Another useful vector is Fe, the equilibrant of Fx and Fy. Fe is the force needed to exactly
offset the combined pull of the two ships. Fe has the same magnitude as Fr, but is in the
opposite direction. As you will see in the following experiment, the equilibrant provides a useful
experimental method for finding the resultant of two or more forces.
Setup
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Set up the equipment as shown in Figure 2.2. The Mass Hanger and mass provide a gravitational
force of F = mg downward. However, since the Force Ring is not accelerated, the downward
force must be exactly balanced by an equal and opposite, or equilibrant, force. This equilibrant
force, Fe, is of course provided by the Spring Balance.
Procedure
À What is the magnitude and direction of F, the gravitational force provided by the mass and
Mass Hanger (F = mg)?
F: Magnitude = _______________ .
Direction = ___________________ .
Use the Spring Balance and the Degree Plate to determine the magnitude and direction of Fe.
Fe: Magnitude = __________________ .
Direction = ___________________ .
Now use pulleys and hanging masses as shown in Figure 2.3 to set up the equipment so that two
known forces, F1 and F2, are pulling on the Force Ring. Use the Holding Pin to prevent the ring
from being accelerated. The Holding Pin provides a force, Fe, that is exactly opposite to the
resultant of F1 and F2.
Adjust the Spring Balance to determine the magnitude of Fe. As shown, keep the Spring
Balance vertical and use a pulley to direct the force from the spring in the desired direction.
Move the Spring Balance toward or away from the pulley to vary the magnitude of the force.
Adjust the pulley and Spring Balance so that the Holding Pin is centered in the Force Ring.
NOTE: To minimize the effects of friction in the pulleys, tap as needed on the Experiment
Board each time you reposition any component. This will help the Force Ring come to its true
equilibrium position.
Record the magnitude in newtons of F1, F2, and Fe; the value of the hanging masses, M1, and
M2
(include the mass of the mass hangers); and alsoq1, q2, and qe, the angle each vector makes
with
respect to the zero-degree line on the degree scale.
M1 = _________(kg)__
F1
: (Magnitude) = __________(N)____ Angle = ____________
M2 = __________(kg) F2: Magnitude = ______________ Angle = ____________
Fe: Magnitude = ______________ Angle = ____________
Use the values you recorded above to construct F1, F2, and Fe on a separate sheet of paper.
Choose an appropriate scale (such as 2.0 cm/newton) and make the length of each vector
proportional to the magnitude of the force. Label each vector and indicate the magnitude of the
force it represents.
On your diagram, use the parallelogram method to draw the resultant of F1 and F2. Label the
resultant Fr. Measure the length of Fr to determine the magnitude of the resultant force and
record this magnitude on your diagram.
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Does the equilibrant force vector, Fe, exactly balance the resultant vector, Fr. If not, can you
suggest some possible sources of error in your measurements and constructions? Vary the
magnitudes and directions of F1 and F2 and repeat the experiment.
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Experiment 3: Resolving Forces—Components
EQUIPMENT NEEDED:
– Experiment Board – Degree Scale
– Force Ring – Pulleys (3)
– Mass Hangers (3) – Masses
– String
Theory
In Experiment 2, you added concurrent forces vectorially to determine the magnitude and
direction of the combined force. In this experiment, you will do the opposite; you will find two
forces which, when added together, have the same effect as the original force. As you will see,
any force vector in the x-y plane can be expressed as the sum of a vector in the x direction and a
vector in the y direction.
Set Up.
As shown, determine a force vector, F, by hanging a mass from the Force Ring over a pulley.
Use the Holding Pin to hold the Force Ring in place. Set up the Spring Balance and a pulley so
the string from the balance runs horizontally from the bottom of the pulley to the Force Ring.
Hang a second Mass Hanger directly from the Force Ring. Now pull the Spring Balance toward
or away from the pulley to adjust the horizontal, or “xcomponent” of the force. Adjust the mass
on the vertical Mass Hanger to adjust the vertical or “y-component” of the force. Adjust the x
and y components in this way until the Holding Pin is centered in the Force Ring. (Notice that
these x and y components are actually the x and y components of the equilibrant of F, rather
than of F itself.)
NOTE: The hanging masses allow the mass to be varied only in 10 gram increments. Using an
additional Mass Hanger as a mass allows adjustments in 5 gram increments. Paper clips are
convenient for more precise variation. Weigh a known number of clips with the Spring Balance
to determine the mass per clip.
Procedure
Record the magnitude and angle of F. Measure the angle as shown in Figure 3.1.
Magnitude =_________________ Angle = _______________ .
Record the magnitude of the x and y components of the equilibrant of F.
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x-Component = ______________ y-Component = _____________ .
What are the magnitudes of Fx and Fy, the x and y components of F?
Fx = ______________ Fy =______________ .
Change the magnitude and direction of F and repeat the experiment.
Record the angle of F, and the magnitudes of F, Fx, and Fy.
F: Magnitude = ______________ Angle = _______________ .
Fx = _______________________ Fy = __________________
Why use components to specify vectors? One reason is that using components makes it easy to
add vectors mathematically. Figure 3.2 shows the x and y components of a vector of length F, at
an angle q with the x-axis. Since the components are at right angles to each other, the
parallelogram used to determine their resultant is a rectangle.
Using right triangle AOX, the components of F are easily calculated: the x-component equals F
cos q; the y-component equals F sin q. If you have many vectors to add, simply determine the x
and y components for each vector.
Add all the x-components together and add all the y-components together. The resulting values
are the x and y components for the resultant.
Set up the equipment as in the first part of this experiment, using a pulley and a hanging mass to
establish the magnitude and direction of a force vector. Be sure the x-axis of the Degree Plate is
horizontal
Record the magnitude and angle of the force vector, F, that you have constructed.
Magnitude =_________________ Angle = _______________ .
Calculate Fx and Fy, the magnitudes of the x and y components of
F (Fx = F cos q; Fy = F sin q).
Fx = _______________________ Fy = __________________
Now set up the Spring Balance and a hanging mass, as in the first part of this experiment
(Figure 3.1). Using the values you calculated in question 6, position the Spring Balance so it
pulls the Force Ring horizontally by an amount Fx. Adjust the hanging mass so it pulls the
Force Ring vertically down by an amount Fy.
Questions
À Is the Force Ring at equilibrium in the center of the Degree Plate? Generally it is most useful
to find the components of a vector along two perpendicular axes, as you did above. However, it
is not necessary that the x and y axes be perpendicular. If time permits, try setting up the
equipment to find the components of a vector along nonperpendicular axes. (Use pulleys to
redirect the component forces to non-perpendicular directions.)
24
What difficulties do you encounter in trying to adjust the x and y components to resolve a vector
along non-perpendicular axes?
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27
Experiment 4: Torque—Parallel Forces
EQUIPMENT NEEDED:
– Experiment Board – Balance Beam
– Pivot – Mass Hangers (3)
– Masses – Tape
Theory
In Experiment 2, you found resultants and equilibrants for concurrent forces—forces that act
upon the same point. In the real world, however, forces are often not concurrent. In Figure 4.1,
for example, two spaceships are pulling on different points of an asteroid. Two questions might
be asked: À Which direction will the asteroid be accelerated in?
Will the asteroid rotate?
If both tow ropes were attached to point A, the resultant would be the force vector shown, Fr. In
fact, Fr does point in the direction in which the asteroid will be accelerated (this idea will be
investigated further in later experiments).
However, what of question 2. Will the asteroid rotate? In this experiment you will begin to
investigate the types of forces that cause rotation in physical bodies. In doing so, you will
encounter a new concept—torque.
28
Setup
Using a magic marker pen, draw a horizontal line on the Experiment Board. (The Inclined Plane
can be used as a level and a straightedge to ensure a truly horizontal line.) Then set up the
equipment as shown in Figure
4.2. Adjust the beam in the pivot retainer until the beam is perfectly balanced on the pivot. Use
your horizontal line as a reference.
NOTE: To prevent the pivot point from sliding, place a piece of thin tape against either edge of
the pivot retainer. Add additional small pieces of tape, if needed, to rebalance the beam.
Slide a plastic retainer with a hook onto each end of the beam, then hang a Mass Hanger from
each hook as shown in Figure 4.3. Position one Mass Hanger approximately half way between
the pivot point and the end of the beam. Slide the other Mass Hanger on the beam until the beam
is perfectly balanced.
Procedure
À Measure d1 and d2, the distances of each Mass Hanger from the pivot point (see Figure 4.3).
d1 = ___________________
d2 = ___________________ .
Add a 50-gram mass to each Mass Hanger. Á Is the beam still balanced? Add an additional 20-
gram mass to one Mass Hanger. Â Can you restore the balance of the beam by repositioning the
hanger? Place 75 grams of mass on one Mass Hanger (M1 in Figure 4.4) and position it
approximately half way between the pivot and the end of the beam, as shown.
Place various masses on the other Mass Hanger (M2), and slide it along the beam as needed to
rebalance the beam. At each balanced position, measure the mass (M) hanging from each Mass
Hanger and the distance (d) between the Mass Hanger and the pivot point, as shown in the
illustration. Take measurements for at least 5 different values of M2 and record your results in
Table 4.1. Be sure to include the units of your measurements. Vary M1 and repeat your
measurements.
NOTE: For accurate results, include the mass of the Mass Hangers (5 grams) and of the plastic
retainers and hooks (2.2 grams) when determining M1 and M2.
Use the formula F = Mg, where g is the acceleration due to gravity, to determine the
gravitational forced produced by the hanging masses in each case. Then perform the
calculations shown in the table to determine t1 and t2; that is,
1 = F1 d1, and 2 = F2
d2. Record your calculated values for each balanced
position of the beam.
29
The quantity 1 = F1 d1 is called the torque of the force
F1 about the pivot point of the balance beam.
Questions
1. From your results, what mathematical relationship must hold between t1 and t2 in order for the
beam to be balanced? There is an additional force on the beam, besides F1 and F2. The pivot
pulls up on the beam—otherwise, of course, the beam would accelerate downward.
2. What torque is produced about the pivot point of the balance beam by the upward pull of the
pivot? If you have time, try adding a third Mass Hanger with masses (M3) to the balance beam,
on the same side of the beam as M2.
3. What relationship must hold between t1 , t2 , and t3 in order for the beam to be balanced?
30
31
32
Experiment 5: Center of Mass
EQUIPMENT NEEDED:
– Experiment Board – Pivot
– Planar Mass – Balance Beam
– Mass Hanger (1) – 50-gram Mass
– String
Theory
Gravity is a universal force; every bit of matter in the universe is attracted to every other bit of
matter. So when the balance beam is suspended from a pivot point, every bit of matter in the
beam is attracted to every bit of matter in the Earth.
Fortunately for engineers and physics students, the sum of all these gravitational forces
produces a single resultant. This resultant acts as if it were pulling between the center of the
Earth and the center of mass of the balance beam. The magnitude of the force is the same as if
all the matter of the Earth were located at the center of the Earth, and all the matter of the
balance beam were located at the center of mass of the balance beam. In this experiment, you
will use your understanding of torque to understand and locate the center of mass of an object.
Setup
Hang the Balance Beam from the pivot as shown in Figure 6.1. As in Experiment 4, use the
Inclined Plane as a level and straightedge to draw a horizontal reference line. Adjust the position
of the Balance Beam in the pivot so that the beam balances horizontally.
Since the Balance Beam is not accelerated, the force at the pivot point must be the equilibrant of
the total gravitational force acting on the beam. Since the beam does not rotate, the gravitational
force and its equilibrant must be concurrent forces.
Experiment
À Why would the Balance Beam necessarily rotate if the resultant of the gravitational forces
and the force acting through the pivot were not concurrent forces? Think of the Balance Beam
as a collection of many small hanging masses. Each hanging mass is pulled down by gravity and
therefore provides a torque about the pivot point of the Balance Beam.
33
What is the relationship between the sum of the clockwise torques about the center of mass and
the sum of the counterclockwise torques about the center of mass? Explain.
On the basis of your answer to question 1, use a pencil to mark the center of mass of the balance
beam. Then attach a Mass Hanger to each end of the beam. Hang 50 grams from one hanger,
and 100 grams from the other, as shown in Figure 6.2. Now slide the beam through the pivot
retainer until the beam and masses are balanced and the beam is horizontal. The pivot is now
supporting the beam at the center of mass of the combined system (i.e. balance beam plus
hanging masses).
Calculate the torques, t1, t2, and t3, provided by the forces F1, F2, and F3 acting about the new
pivot point, as shown in the illustration. Be sure to indicate whether each torque is clockwise
(cw) or counterclockwise (ccw).
1 = ________________2 =_______________3 = _______________ .
Are the clockwise and counterclockwise torques balanced? Remove the 50 gram mass and Mass
Hanger. Reposition the beam in the pivot to relevel the beam. Recalculate the torques about the
pivot point. Are the torques balanced?
Hang the Planar Mass from the Holding Pin of the Degree Plate as shown in Figure 6.3. Since
the force of the Pin acting on the mass is equilibrant to the sum of the gravitational forces acting
on the mass, the line of the force exerted by the Pin must pass through the center of mass of the
Planar Mass. Hang a piece of string with a hanging mass from the Holding Pin. Tape a piece of
paper to the Planar Mass as shown. Mark the paper to indicate the line of the string across the
Planar Mass. Now hang the Planar Mass from a different point. Again, mark the line of the
string. By finding the intersection of the two lines, locate the center of mass of the Planar Mass.
Hang the Planar Mass from a third point.
Does the line of the string pass through the center of mass?
Would this method work for a three dimensional object? Why or why not?
34
35
Experiment 6: Sliding Friction
EQUIPMENT NEEDED:
– Experiment Board – Inclined Plane
– Friction Block – Spring Balance
– Pulley (1) – Mass Hangers (2)
– Masses – String
Theory
In most physical systems, the effects of friction are not easily predicted, or even measured. The
interactions between objects that cause them to resist sliding against each other seem to be due
in part to microscopic irregularities of the surfaces, but also in part to interactions on a
molecular level. However, though the phenomena is not fully understood, there are some
properties of friction that hold for most materials under many different conditions. In this
experiment you will investigate some of the properties of sliding friction the force that resists
the sliding motion of two objects when they are already in motion.
Procedure
Use the Spring Scale to determine W, the magnitude of the weight of the Friction Block.
W = __________________ .
Set up the equipment as shown in Figure 9.1. Use the built-in plumb bob to ensure that the
Inclined Plane is level. Adjust the position of the pulley so that the string is level with the
surface of the Inclined Plane. Adjust the mass on the Mass Hanger until, when you give the
Friction Block a small push to start it moving, it continues to move along the Inclined Plane at a
very slow, constant speed.
If the block stops, the hanging mass is too light; if it accelerates, the mass is too large. The
weight of the hanging mass that is just sufficient to provide a constant slow speed is Ff, the
force of the sliding friction of the Friction Block against the Inclined Plane.
Three variables can be varied while measuring Ff.
They are: Normal Force (W + Mg)—Place masses of weight W on top of the Friction Block to
adjust
the normal force between the block and the Inclined Plane.
36
Contact Material—Using sides A and B of the Friction Block, wood is the material in contact
with the Inclined Plane. Using side C, only the two strips of teflon tape contact the
Inclined Plane.
Contact Area (A, B, C)—Adjust the area of contact between the Friction Block by having side
A, B, or C of the Friction Block in contact with the Inclined Plane. (NOTE: Using side C, the
contact area is the surface area of the two strips of teflon tape.)
Adjust the mass on top of the Friction Block to each of the values shown in Table 9.1. At each
value of M, adjust the hanging mass to determine the magnitude of Ff. Perform this
measurement using side A, B, and C of the Friction Block. For each measurement, calculate the
ratio between the magnitude of the sliding friction (Ff) and the magnitude of the normal force
(W + Mg). This ratio is called the coefficient of friction, m.
NOTE: You will need to adjust the hanging mass in small increments. Paper clips are
convenient for this purpose. Weigh a large number of paper clips on the Spring Balance and
divide by the number of clips to determine the weight per clip.
Based on your measurements:
Questions
1. Does the value of sliding friction between two objects depend on the normal force between the
two objects? If so, what is the relationship between normal force and sliding friction?
2. Does the value of sliding friction between two objects depend on the area of contact between the
two objects?
3. Does the value of sliding friction between two objects depend on the materials that are in
contact?
M (mass
of block)
(kg)*10-3
(block+other
weights)
W + Mg (N)
Ff (N) = hanging mass
(mg)
µ = (mg)/ (W+Mg)
A C A C
0
50
100
150
200
250
37
38
39
Experiment 7: Simple Harmonic Motion: Mass on a Spring
EQUIPMENT NEEDED:
– Experiment Board – Spring Balance
– Mass Hanger – Masses
– Stopwatch
Theory
Figure 10.1 shows a mass hanging from a spring. At rest, the mass hangs in a position such that
the spring force just balances the gravitational force on the mass. When the mass is below this
point, the spring pulls it back up. When the mass is above this point, gravity pulls it back down.
The net force on the mass is therefore a restoring force, because it always acts to accelerate the
mass back toward its equilibrium position.
In Experiment 1 you investigated Hook’s Law, which states that the force exerted by a spring is
proportional to the distance beyond its normal length to which it is stretched (this also holds true
for the compression of a spring).
This idea is stated more succinctly in the mathematical relationship: F = -kx;
where F is the force exerted by the spring, x is the displacement of the end of the spring from its
equilibrium position, and k is the constant of proportionality, called the spring constant (see
Experiment
1). Whenever an object is acted on by a restoring force that is proportional to the displacement
of the object from its equilibrium position, the resulting motion is called Simple Harmonic
Motion. When the simple harmonic motion of a mass (M) on a spring is analyzed
mathematically using Newton’s Second Law (the analysis requires calculus, so it will not be
shown here), the period of the motion (T) is found to be:
40
In this experiment, you will experimentally test the validity of this equation.
Experiment
À Measure k, the spring constant for the spring in the Spring Balance (see Experiment 1).
k = _______________(newtons/meter).
Set up the equipment as shown in Figure 10.1, with 120 grams on the Mass Hanger (125 grams
total mass, including the hanger). Be sure that the Spring Balance is vertical so that the rod
hangs straight down through the hole in the bottom of the balance. This is important to
minimize friction against the side as the mass oscillates.
Now pull the rod down a few centimeters. Steady the mass, then let go of the rod. Practice until
you can release the rod smoothly, so that the mass and the rod oscillate up and down and there is
no rubbing of the rod against the side of the hole.
Set the mass oscillating. Measure the time it takes for at least 10 full oscillations to occur.
(Measure the time for as many oscillations as can be conveniently counted before the amplitude
of the oscillations becomes too small.) Record the mass, the time, and the number of oscillations
counted in Table 10.1. Divide the total time by the number of oscillations observed to determine
the period of the oscillations. (The period is the time required for one complete oscillation).
Record this value in the table.
Repeat the measurement 5 times. Calculate the period for each measurement. Then add your
five period measurements together and divide by 5 to determine the average period over all five
measurements.
Use the equation given at the beginning of this Experiment to calculate a theoretical value for
the period using each mass value. (Since the spring constant is in units of newtons/m, your mass
values used in the equation must be in kg.) Enter this value in the table.
1. Does your theoretical value for the period accurately predict your experimental value? Repeat
the experiment using masses of 175 and 225 grams (including the mass of the Mass Hanger).
41
2. Does the equation for the period of an oscillating mass provide a good mathematical model for
the physical reality?
42
43
Experiment 8: Simple Harmonic Motion—the Pendulum
EQUIPMENT NEEDED:
– Experiment Board – Pivot
– Mass Hanger – Masses
– String
Theory
Simple harmonic motion is not restricted to masses on springs. In fact, it is one of the most
common and important types of motion found in nature. From the vibrations of atoms to the
vibrations of airplane wings, simple harmonic motion plays an important role in many physical
phenomena. A swinging pendulum, for example, exhibits behavior very similar to that of a mass
on a spring. By making some comparisons between these two phenomena, some predictions can
be made about the period of oscillations for a pendulum.
Figure 11.1 shows a pendulum with the string and mass at an angle q from the vertical position.
Two forces act on the mass; the force of the string and the force of gravity. The gravitational
force, F = mg, can be resolved into two components; Fx and Fy. Fy just balances the force of
the string and therefore does not accelerate the mass. Fx is in the direction of motion of the
mass, and therefore does accelerate and decelerate the mass.
Using the two congruent triangles in the diagram, it can be seen that Fx = mg sinq, and that the
displacement of the mass from its equilibrium position is an arc whose distance, x, is
approximately L tanq. If the angle q is reasonably small, then it is very nearly true that sinq =
tanq. Therefore, for small swings of the pendulum, it is approximately true that Fx = mgtanq =
mgx/L. (Since Fx is a restoring force, the equation could be stated more accurately as Fx = -
mgx/L.) Comparing this equation with the equation for a mass on a spring (F = -kx), it can be
seen that the quantity mg/L plays the same mathematical role as the spring constant. On the
basis of this similarity, you might speculate that the period of motion for a pendulum is just:
g
LT 2 ; g=
2
24
T
L
where m is the mass, g is the acceleration due to gravity, and L is distance from the pivot point
to the center of mass of the hanging mass. In this experiment, you will test the validity of this
equation.
44
L (m) t (s)
(10 T)
Period (s)
T =t/10
(g’)
Experiment
Hang a Mass Hanger from the pivot as shown in Figure 11.1. Set the mass swinging, but keep
the angle of the swing reasonably small. Measure the time it takes for at least 30 full oscillations
to occur. In Table 11.1, record the mass, the distance L, the time, and the number of oscillations
counted. Divide the total time by the number of oscillations observed to determine the period of
the oscillations. (The period is the time required for one complete oscillation). Record this value
in the table. Repeat the measurement 5 times. Calculate the period for each measurement. Then
add your five period measurements together and divide by 5 to determine the average period
over all five measurements Repeat your measurements using a different mass.
Use the equation given at the beginning of this Experiment to calculate a theoretical value for
the period in each case (g = 9.8 N/m; be sure to express L in meters when you plug into the
equation). Enter this value in the table.
Does the period of the oscillations depend on the mass of the pendulum?
Does your theoretical value for the period accurately predict your experimental value? Repeat
the experiment using a significantly different string length.
Does the equation for the period of an oscillating mass provide a good mathematical model for
the physical reality?
M (kg) L (m) Total t (s) T, Period
(s)
M1
M2
M3
45
46
Experiment 9: Conservation of Angular Momentum Using a Point Mass
EQUIPMENT REQUIRED
- Smart Pulley Timer Program
- Rotational Inertia Accessory (ME-8953)
- Rotating Platform (ME-8951)
- Smart Pulley
- balance
Purpose
A mass rotating in a circle is pulled in to a smaller radius and the new angular speed is predicted
using conservation of angular momentum.
Theory
Angular momentum is conserved when the radius of the circle is changed.
where Ii i is the initial angular speed. So the final rotational
speed is given by:
where is the angular acceleration which is equal to a/r and is the torque caused by the
weight hanging from the thread which is wrapped around the base of the apparatus.
where r is the radius of the cylinder about which the thread is wound and T is the tension in the
thread when the apparatus is rotating.
Applying Newton’s Second Law for the hanging mass, m, gives (See Figure 4.1)
Solving for the tension in the thread gives:
Once the linear acceleration of the mass (m) is determined, the torque and the angular
acceleration
can be obtained for the calculation of the rotational inertia
47
48
Experiment 10: Rotational Inertia of Disk and Ring
EQUIPMENT REQUIRED
- Precision Timer Program - mass and hanger set
- Rotational Inertia Accessory (ME-9341) - paper clips (for masses < 1 g)
- Smart Pulley - triple beam balance
- calipers
Purpose
The purpose of this experiment is to find the rotational inertia of a ring and a disk
experimentally and to verify that these values correspond to the calculated theoretical values.
Theory
Theoretically, the rotational inertia, I, of a ring about its center of mass is given by:
where M is the mass of the ring, R1 is the inner radius of the ring, and R2 is the outer radius of
the ring. See Figure 5.1.
The rotational inertia of a disk about its center of mass is given by:
where M is the mass of the disk and R is the radius of the disk. The rotational inertia of a disk
about its diameter is given by:
49
To find the rotational inertia experimentally, a known torque is applied to the object and the
resulting angular acceleration is measured. Since = I
where is the angular acceleration which is equal to a/r and is the torque caused by the
weight
hanging from the thread which is wrapped around the base of the apparatus.
= rT
where r is the radius of the cylinder about which the thread is wound and T is the tension in the
thread when the apparatus is rotating.
Applying Newton’s Second Law for the hanging mass, m, gives (See Figure 5.3)
Solving for the tension in the thread gives:
T = m g – a
Once the linear acceleration of the mass (m) is determined, the torque and the angular
acceleration can be obtained for the calculation of the rotational inertia
Setup
Remove the track from the Rotating Platform and place the disk directly on the center shaft as
shown in Figure 5.4. The side of the disk that has the indentation for the ring should be up.
Place the ring on the disk, seating it in this indentation.
Mount the Smart Pulley to the base and connect it to a computer.
Run the Smart Pulley Timer program.
50
Procedure
Measurements for the Theoretical Rotational Inertia
Weigh the ring and disk to find their masses and record these masses in Table 5.1.
Measure the inside and outside diameters of the ring and calculate the radii R1 and R2. Record
in
Table 5.1.
Measure the diameter of the disk and calculate the radius R and record it in Table 5.1.
Measurements for the Experimental Method
Accounting For Friction
Because the theory used to find the rotational inertia experimentally does not include friction, it
will be compensated for in this experiment by finding out how much mass over the pulley it
takes to overcome kinetic friction and allow the mass to drop at a constant speed. Then this
“friction mass” will be subtracted from the mass used to accelerate the ring.
To find the mass required to overcome kinetic friction run “Display Velocity”: <V>-Display
Velocity <RETURN>; <A>-Smart Pulley/Linear String <RETURN>; <N>-Normal Display
<RETURN>.
51
Put just enough mass hanging over the pulley so that the velocity is constant to three significant
figures. Then press <RETURN> to stop displaying the velocity. Record the friction mass in
Table 5.2.
Finding the Acceleration of Ring and Disk
To find the acceleration, put about 50 g over the pulley and run “Motion Timer”: <M>-Motion
Timer <RETURN> Wind the thread up and let the mass fall from the table to the floor, hitting
<RETURN> just before the mass hits the floor.
Wait for the computer to calculate the times and then press <RETURN>. To find the
acceleration,
graph velocity versus time: <G>-Graph Data <RETURN>; <A>-Smart Pulley/Linear String
<RETURN;> <V>-Velocity vs. Time <R>-Linear Regression <SPACEBAR> (toggles it
on) <S>-Statistics <SPACEBAR> <RETURN>.
The graph will now be plotted and the slope = m will be displayed at the top of the graph. This
slope is the acceleration. Record in Table 5.2.
Push <RETURN> and <X> twice to return to the Main Menu.
Measure the Radius
Using calipers, measure the diameter of the cylinder about which the thread is wrapped and
calculate the radius. Record in Table 5.2.
Finding the Acceleration of the Disk Alone
Since in Finding the Acceleration of Ring and Disk the disk is rotating as well as the ring, it is
necessary to determine the acceleration, and the rotational inertia, of the disk by itself so this
rotational inertia can be subtracted from the total, leaving only the rotational inertia of the ring.
To do this, take the ring off the rotational apparatus and repeat Finding the Acceleration of
Ring and Disk for the disk alone.
NOTE: that it will take less “friction mass” to overcome the new kinetic friction and it is only
necessary to put about 30 g over the pulley in Finding the Acceleration of Ring and Disk.
Disk Rotating on an Axis Through Its Diameter
Remove the disk from the shaft and rotate it up on its side. Mount the disk vertically by
inserting the shaft in one of the two “D”-shaped holes on the edge of the disk. See Figure 5.5.
52
WARNING! Never mount the disk vertically using the adapter on the track. The adapter is too
short for this purpose and the disk might fall over while being rotated.
Repeat steps Measure the Radius and Finding the Acceleration of the Disk Alone to
determine
the rotational inertia of the disk about its diameter. Record the data in Table 5.2.
Calculations
1. Record the results of the following calculations in Table 5.3.
2. Subtract the “friction mass” from the hanging mass used to accelerate the apparatus to
determine
3. the mass, m, to be used in the equations.
4. Calculate the experimental value of the rotational inertia of the ring and disk together.
5. Calculate the experimental value of the rotational inertia of the disk alone.
6. Subtract the rotational inertia of the disk from the total rotational inertia of the ring and disk.
This will be the rotational inertia of the ring alone.
7. Calculate the experimental value of the rotational inertia of the disk about its diameter.
8. Calculate the theoretical value of the rotational inertia of the ring.
9. Calculate the theoretical value of the rotational inertia of the disk about its center of mass and
about its diameter.
10. Use a percent difference to compare the experimental values to the theoretical values.
53
54
55
Deney 11 : Eğik Atış Rotası
GEREKLİ EKİPMANLAR:
- Eğik atıcı ve Plastik Top - Ölçme şeridi ya da cetvel
- Karbon Kağıdı - Beyaz Kağıt
- Hareket edebilir düşey hedef tahtası (Yerden namluya kadar erişebilmeli)
- Grafik Kağıdı
Amaç
Bu deneyin amacı: bir cisim bir yükseklikten yatay olarak fırlatıldığında, düştüğü düşey
mesafenin, topun hareket ettiği yatay mesafeye nasıl bağlı olduğunun bulunmasıdır.
Teori
Menzil, Fırlatıcının namlusu ile topun vurduğu hedef arasındaki yatay x mesafesidir, verilen x
= v0t denkleminde v0 topun namluyu terk ettiği andaki başlangıç hızı ve t uçuş süresidir.
Eğer top yatay olarak fırlatılmışsa, topun uçuş süresi şöyle olacaktır
ov
xt
Düşey uzaklık, y, topun t süresinde yaptığı düşey mesafe şu şekildedir
2
2
1gty
burada g yerçekimine bağlı ivmedir.
y için verilen eşitlikte t' yi yerine koyduğumuzda;
2
2
0
xV2
gy
x2 ' ye karşı çizilen y grafiği, eğimi
2.2 ov
g olan düz bir çizgi verir.
56
Kuruluş
Eğik Atıcıyı sağlam bir masanın bir kenarına, atıcı bu kenardan dışarı doğru
fırlatma yapacak şekilde tutturun.
Topun yatay olarak
fırlatılmasını
sağlayacak şekilde,
Eğik Atıcının
açısını sıfır
dereceye ayarlayın.
Düşey hedefin başlan-
gıç pozisyonunu belir-
lemek için orta derece-
de bir mesafeden bir
deneme atışı yapın. Şekil M6.1 Kuruluş
Top hedefi yere yakın bir yerde vuracak şekilde hedefi yerleştirin. Şekil M6.1.
Hedef tahtasını beyaz kağıtla kaplayın. Beyaz kağıdın üzerine karbon kağıdını bantlayın.
Deneyin Yapılışı
Yerden namluya kadar olan düşey yüksekliği ölçün ve Tablo M6.1.'e kaydedin. Bu
yüksekliği hedef üzerinde işaretleyin.
Eğik Atıcının namlusundan hedefe olan yatay mesafeyi ölçün ve Tablo M6.1.' e kaydedin.
Topu fırlatın.
Hedefi Firlaticiya 10 ile 20 cm yaklastirin.
2' den 4' e kadar olan aşamaları topun, namlunun yüksekliğinin yaklaşık 10 ile 20 cm
aşağısında ki hedefi vurduğunda ki yüksekliğinde tekrarlayın.
Tablo M6.1 Veriler
Namlunun Yüksekliği = ____________m
(Yatay mesafe) x (m) x² (m2) (Yükseklik) y (m)
Analiz
Hedef üzerinde top tarafindan birakilan izin namlu hizasına göre düşey mesafesini ölçün
ve Tablo M6.1.' e kaydedin.
57
Tüm veri noktalari için x² ' yi hesaplayin ve Tablo M6.1' e kaydedin.
x² ' ye karsi y noktalarini isaretleyin ve bunlardan geçen en uygun doğruyu çizin.
Grafiğin eğimini hesaplayın ve Tablo M6.2' ye kaydedin.
Grafiğin eğiminden, topun namluyu terk ettiği andaki başlangıç hızını hesaplayın ve Tablo
M6.2' ye kaydedin.
x ve y için veri noktalarından, y' yi kullanarak zamanı ve sonra x' i ve zamanı kullanarak
başlangıç hızını hesaplayın. Sonuçları Tablo M6.2.' ye kaydedin.
Bu iki metodu kullanarak bulunan, başlangıç hızları arasındaki farkı fark yüzdesini
hesaplayın. Tablo M6.2.' ye kaydedin.
Tablo M6.2
Grafiğin eğimi
Grafiğin eğiminden bulunan başlangıç hız
(m/s)
İki Fotogate Arasını geçiş süresi, t (s)
Fotogate yardımı ile bulunan başlangıç hız
(m/s)
Fark yüzdesi
Sorular
Eğik atış hareketinin yörünge denklemini çıkarınız. Grafiğini çiziniz.
58
59
Experiment 12: Conservation of Angular Momentum
EQUIPMENT REQUIRED
- Smart Pulley Timer Program - balance
- Rotational Inertia Accessory (ME-8953)
- Rotating Platform (ME-8951)
- Smart Pulley Photogate
Purpose
A non-rotating ring is dropped onto a rotating disk and the final angular speed of the system is
compared with the value predicted using conservation of angular momentum.
Theory
When the ring is dropped onto the rotating disk, there is no net torque on the system since the
torque on the ring is equal and opposite to the torque on the disk. Therefore, there is no change
in angular momentum. Angular momentum is conserved.
where Ii
i is the initial angular speed. The initial rotational
inertia is that of a disk
and the final rotational inertia of the combined disk and ring is
If = (1/2)M1R2 + (1/2)M2(r1
2+r2
2)
So the final rotational speed is given by
Setup
Level the apparatus using the square mass on the track
60
Assemble the Rotational Inertia Accessory as shown in Figure 7.1. The side of the disk with the
indentation for the ring should be up.
Mount the Smart Pulley photogate on the black rod on the base and position it so it straddles the
holes in the pulley on the center rotating shaft.
Run the Smart Pulley Timer program.
Procedure
Select <M>-Motion Timer <RETURN>.
Hold the ring just above the center of the disk. Give the disk a spin using your hand. After about
25 data points have been taken, drop the ring onto the spinning disk See Figure 7.2.
Continue to take data after the collision and then push <RETURN> to stop the timing.
When the computer finishes calculating the times, graph the rotational speed versus time. <A>-
Data Analysis Options <RETURN> <G>-Graph Data <RETURN> <E>-Rotational Apparatus
<RETURN> <V>-Velocity vs. Time <RETURN>
After viewing the graph, press <RETURN> and choose <T> to see the table of the angular
velocities. Determine the angular velocity immediately before and immediately after the
collision.
Record these values in Table 7.1.
Weigh the disk and ring and measure the radii. Record these values in Table 7.1.
Analysis
Calculate the expected (theoretical) value for the final angular velocity and record this value in
Table 7.1.
Calculate the percent difference between the experimental and the theoretical values of the final
angular velocity and record in Table 7.1.
61
Questions
Does the experimental result for the angular speed agree with the theory?
What percentage of the rotational kinetic energy lost during the collision? Calculate this and
record the results in Table 7.1.
62
63
64
Experiment 13: Projectile Motion
Equipment Needed
Item Item
Projectile Launcher and plastic ball Plumb bob and string
Meter stick Carbon paper
White paper Sticky tape
Purpose
The purpose of this experiment is to predict and verify the range of a ball launched at an angle.
The initial speed of the ball is determined by shooting it horizontally and measuring the range of
the ball and the height of the Launcher.
Theory
To predict where a ball will land on the floor when it is shot from the Launcher at some angle
above the horizontal, it is first necessary to determine the initial speed (muzzle velocity) of the
ball. That can be determined by shooting the ball horizontally from the Launcher and measuring
the vertical and horizontal distances that the ball travels.
The initial speed can be ued to calculate where the ball will land when the ball is shot at an
angle above the horizontal.
Initial Horizontal Speed
For a ball shot horizontally with an initial speed, v0, the horizontal distance travelled by the ball
is given by
x = v0t,
where t is the time the ball is in the air. (Neglect air friction.)
The vertical distance of the ball is the distance it drops in time t given by .
The initial speed can by determined by measuring x and y. The time of flight, t, of the ball can
be found using and the initial horizontal speed can be found using .
65
Initial Speed at an Angle
To predict the horizontal range, x, of a ball shot with an initial speed, v0, at an angle, , above
the horizontal, first predict the time of flight from the equation for the vertical motion:
where y0 is the initial height of the ball and y is the position of the ball when it hits the floor. In
other words, solve the quadratic equation for t and then use x = v0 cost where v0 cos is the
horizontal component of the initial speed.
Setup
1. Clamp the Projectile Launcher to a sturdy table or other horizontal surface. Mount the
Launcher near one end of the table.
2. Adjust the angle of the Projectile Launcher to zero degrees so the ball will by launched
horizontally.
Item Item
Projectile Launcher and plastic ball Plumb bob and string Meter stick Carbon paper White paper
Sticky
Part A: Determining the Initial Horizontal Speed of the Ball
1. Put a plastic ball in the Projectile Launcher and use the ramrod to cock it at the long range
position. Fire one shot to locate where the ball hits the floor. At that point, tape a piece of white
paer to the floor. Place a piece of carbon paper (carbon-side down) on top of the white paper
and tape it in place. • When the ball hits the carbon paper on the floor, it will leave a mark on
the white paper.
2. Fire ten shots.
3. Measure the vertical distance from the bottom of the ball as it leaves the barrel to the floor.
Record this distance in the Data Table.
The “Launch Position of Ball” in the barrel is marked on the label on the side of the Launcher.
4. Use a plumb bob to find the point on the floor that is directly beneath the release point on the
barrel. Measure the horizontal distance along the floor from the release point to the leading edge
of the piece of white paper. Record the distance in the Data Table.
5. Carefully remove the carbon paper and measure from the leading edge of the White paper to
each of the ten dots. Record these distances in the Data Table and find the average. Calculate
and record the total horizontal distance (distance to paper plus average distance from edge of
paper to dots).
6. Using the vertical distance, y, and the total horizontal distance, x, calculate the time of flight,
t, and the initial horizontal speed of the ball, v0. Record the time and speed in the Data Table.
Part B: Predicting the Range of a Ball Shot at an Angle
1. Adjust the angle of the Projectile Launcher to an angle between 30 and 60 degrees. Record
this angle in the second Data Table.
2. Using the initial speed and vertical distance from the first part of this experiment, calculate
the new time of flight and the new horizontal distance based on the assumption that the ball is
shot at the new angle you have just selected. Record the predictions in the second Data Table.
3. Draw a line across the middle of a white piece of paper and tape the paper on the floor so that
the line on the paper is at the predicted horizontal distance from the Projectile Launcher. Cover
the white paper with carbon paper (carbon side down) and tape the carbon paper in place.
4. Shoot the ball ten times.
5. Carefully remove the carbon paper. Measure the distances to the ten dots and record the
distances in the second Data Table.
66
Data Table A: Determine the Initial Speed
Vertical distance = _____________ Horizontal distance to edge of paper = _________________
Calculated time of flight = ________________ Initial speed = _____________
Data Table B: Predict the Range
Angle above horizontal = _____________ Horizontal distance to edge of paper =
_________________
Calculated time of flight = ________________ Predicted range = _____________
67
68
69
Experiment 14: Projectile Motion Using Photogates
Equipment Needed
Item Item
Projectile Launcher and plastic ball Plumb bob and string
Photogate Head ME-9498A (2) Photogate Mounting Bracket ME-6821A
PASCO Interface or Timer* PASCO Data acquisition software*
Meter stick Carbon paper
White paper Sticky tape
Purpose
The purpose of this experiment is to predict and verify the range of a ball launched at an angle.
Photogates are used to determine the initial speed of the ball.
Theory
To predict where a ball will land on the floor when it is shot from the Launcher at some angle
above the horizontal, it is first necessary to determine the initial speed (muzzle velocity) of the
ball. The speed can be determined by shooting the ball and measuring a time using photogates.
To predict the range, x, of the ball when it is shot with an initial speed at an angle, , above the
horizontal, first predict the time of flight using the equation for the vertical motion:
where y0 is the initial height of the ball and y is the position of the ball when it hits the floor.
Solve the quadratic equation to find the time, t. Use x = (v0 cos t) to predict the range.
Setup
1. Clamp the Projectile Launcher to a sturdy table or other horizontal surface. Mount the
Launcher near one end of the table.
2. Adjust the angle of the Projectile Launcher to an angle between 30 and 60 degrees and record
the angle.
3. Attach the photogate mounting bracket to the Launcher and attach two photogates to the
bracket. Check that the distance between the photogates is 0.10 m (10 cm).
4. Plug the photogates into an interface or a timer.
Procedure
Part A: Determining the Initial Speed of the Ball
1. Put a plastic ball in the Projectile Launcher and use the ramrod to cock it at the long range
position.
2. Setup the data acquisition software or the timer to measure the time between the ball blocking
the two photogates.
3. Shoot the ball three times and calculate the average of these times. Record the data in the
Data Table.
4. Calculate the initial speed of the ball based on the 0.10 m distance between the photogates.
Record the value.
70
Part B: Predicting the Range of a Ball Shot at an Angle
1. Keep the angle of the Projectile Launcher at the original angle above horizontal.
2. Measure the vertical distance from the bottom of the ball as it leaves the barrel to the
floor. Record this distance in the second Data Table.
• The “Launch Position of Ball” in the barrel is marked on the label on the side of the
Launcher.
3. Use the vertical distance, the angle, and the initial speed to calculate the time of flight.
Record the value.
4. Use the time of flight, t, angle, , and initial speed, v0, to predict the horizontal distance
(range, x = (v0 cos t). Record the predicted range.
5. Draw a line across the middle of a white piece of paper and tape the paper on the floor so the
line is at the predicted horizontal distance. Cover the white paper with carbon paper and tape the
carbon paper in place.
6. Use a plumb bob to find the point on the floor that is directly beneath the release point on the
barrel. Measure the horizontal distance along the floor from the release point to the leading edge
of the piece of white paper. Record the distance in the Data Table.
7. Shoot the ball ten times.
8. Carefully remove the carbon paper and measure from the leading edge of the white paper to
each of the ten dots. Record these distances in the Data Table and find the average. Calculate
and record the total horizontal distance (distance to paper plus average distance from edge of
paper to dots).
Angle above horizontal = ______________ Horizontal distance to edge of paper =
_______________
Calculated time of flight = _________________ Predicted range = ________________
71
72
Experiment 15: Polarization of Light
EQUIPMENT INCLUDED:
1 Polarization Analyzer OS-8533A
1 Basic Optics Bench (60 cm) OS-8541
1 Aperture Bracket OS-8534
1 Red Diode Laser OS-8525A
1 Light Sensor CI-6504A
1 Rotary Motion Sensor CI-6538
NOT INCLUDED, BUT REQUIRED:
1 ScienceWorkshop 500 Interface CI-6400
1 DataStudio Software CI-6870
INTRODUCTION
Laser light (peak wavelength = 650 nm) is passed through two polarizers. As the second
polarizer (the analyzer) is rotated by hand, the relative light intensity is recorded as a function of
the angle between the axes of polarization of the two polarizers. The angle is obtained using a
Rotary Motion Sensor that is coupled to the polarizer with a drive belt. The plot of light
intensity versus angle can be fitted to the square of the cosine of the angle.
THEORY
A polarizer only allows light which is vibrating in a particular plane to pass through it. This
plane forms the "axis" of polarization. Unpolarized light vibrates in all planes perpendicular to
the direction of propagation. If unpolarized light is incident upon an "ideal" polarizer, only half
of the light intensity will be transmitted through the polarizer.
Resim Kaynak:
http://micro.magnet.fsu.edu/optics/lightandcolor/polarization.html
Figure 1: Light Transmitted through Two Polarizers
73
The transmitted light is polarized in one plane. If this polarized light is incident upon a second
polarizer, the axis of which is oriented such that it is perpendicular to the plane of polarization
of the incident light, no light will be transmitted through the second polarizer. See Fig.1.
However, if the second polarizer is oriented at an angle not perpendicular to the axis of the first
polarizer, there will be some component of the electric field of the polarized light that lies in the
same direction as the axis of the second polarizer, and thus some light will be transmitted
through the second polarizer.
Unpolarized E-field
Eo
E E
Polarizer #1 Polarizer #2
Figure 2: Component of the Electric Field
If the polarized electric field is called E0 after it passes through the first polarizer, the
component, E, after the field passes through the second polarizer which is at an angle with
respect to the first polarizer is E0
square of the electric field, the light intensity transmitted through the second filter is given by
2cosoII (1)
74
THEORY FOR 3 POLARIZERS
Figure 3: Electric Field Transmitted through Three Polarizers
Unpolarized light passes through 3 polarizers (see Fig.3). The first and last polarizers are
oriented at 90o with respect each other. The second polarizer has its polarization axis rotated an
angle
2
from the second polarizer. The intensity after passing through the first polarizer is I1 and the
intensity after passing through the second polarizer, I2 , is given by
2
12 cosII .
The intensity after the third polarizer, I3 , is given by
2coscos
2cos 22
1
2
23 III (2)
Using the trigonometric identity, sinsincoscoscos , gives
sinsin2
sincos2
cos2
cos
. Therefore, since 2sin
2
1sincos ,
)2(sin4
213
II (3)
Because the data acquisition begins where the transmitted intensity through Polarizer 3 is a o
from the an
o45 (4)
75
SET UP
Figure 4: Equipment Separated to Show Components
1. Mount the aperture disk on the aperture bracket holder.
2. Mount the Light Sensor on the Aperture Bracket and plug the Light Sensor into the
interface (See Fig.5).
Figure 5: ScienceWorkshop 500 Interface with Sensors
3. Rotate the aperture disk so the translucent mask covers the opening to the light sensor
(see Fig.6).
Figure 6: Use Translucent Mask
4. Mount the Rotary Motion Sensor on the polarizer bracket. Connect the large pulley on
the Rotary Motion Sensor to the polarizer pulley with the plastic belt (see Fig.7).
5. Plug the Rotary Motion Sensor into the interface (see Fig 5).
76
Figure 7: Rotary Motion Sensor Connected to Polarizer with Belt
6. Place all the components on the Optics Track as shown in Fig.8.
Figure 8: Setup with Components in Position for Experiment
SOFTWARE SET UP
Start DataStudio and open the file called "Polarization".
PROCEDURE FOR 2 POLARIZERS
In the first two procedure steps, the polarizers are aligned to allow the maximum amount of
light through.
1. Since the laser light is already polarized, the first polarizer must be aligned with the laser's axis
of polarization. First remove the holder with the polarizer and Rotary Motion Sensor from the
track. Slide all the components on the track close together and dim the room lights. Click
START and then rotate the polarizer that does not have the Rotary Motion Sensor until the light
intensity on the graph is at its maximum. You may have to use the button in the upper left on
the graph to expand the graph scale while taking data to see the detail.
2. To allow the maximum intensity of light through both polarizers, replace the holder with the
polarizer and Rotary Motion Sensor on the track, press Start, and then rotate polarizer that does
have the Rotary Motion Sensor until the light intensity on the graph is at its maximum (see Fig.
9).
77
Figure 9: Rotate the Polarizer That Has the Rotary Motion Sensor
3. If the maximum exceeds 4.5 V, decrease the gain on the switch on the light sensor. If
the maximum is less than 0.5 V, increase the gain on the switch on the light sensor.
4. Press Start and slowly rotate the polarizer which has the Rotary Motion Sensor
through 180 degrees. Then press Stop.
ANALYSIS
1. Click on the Fit button on the graph. Choose the User-Defined Fit and write an
equation (Acos(x)^2) with constants you can adjust to make the curve fit your data.
2. Try a cos3 4
(
original fit? Does the equation that best fits your data match theory? If not, why not?
78
PROCEDURE FOR 3 POLARIZERS
NOTE: This section is optional if you do not have a third polarizer. Perhaps another lab group
may have one you can share.
1. Now repeat the experiment with 3 polarizers. Place one polarizer on the track and
rotate it until the transmitted light is a maximum.
2. Then place a second polarizer on the
track and rotate it until the light transmitted
through both polarizers is a minimum.
3. Then place a third polarizer on the track
between the first and second polarizers. Rotate it
until the light transmitted through all three
polarizers is a maximum (see Fig.10).
Figure 10: Setup with a Third Polarizer (#2)
between the Rotary Motion Sensor and the Light
Sensor
4. Press Start and record the Intensity vs. angle for 360 degrees as you rotate the third
polarizer that has the Rotary Motion Sensor.
5. Select your data from 2 polarizers and from 3 polarizers. What two things are
different for the Intensity vs. Angle graph for 3 polarizers compared to 2 polarizers?
6. Click on the Fit button and select the User-Defined Fit. Double-click the User-
Defined Fit box on the graph and enter the equation that you had for two polarizers. Then
change the equation until it matches your data for the 3 polarizers.
QUESTIONS
1. For 3 polarizers, what is the angle between the middle polarizer and the first polarizer
to get the maximum transmission through all 3 polarizers? Remember: In the experiment, the
angle of the middle polarizer automatically reads zero when you start taking data but that
doesn't mean the middle polarizer is aligned with the first polarizer.
2. For 3 polarizers, what is the angle between the middle polarizer and the first polarizer
to get the minimum transmission through all 3 polarizers?
79
80
81
Experiment 16. Interference and Diffraction of Light
Equipment:
INCLUDED:
1 Basic Optics Track, 1.2 m OS-8508
1 High Precision Diffraction Slits OS-8453
1 Basic Optics Diode Laser OS-8525A
1 Aperture Bracket OS-8534B
1 Linear Translator OS-8535
1 High Sensitivity Light Sensor PS-2176
1 Rotary Motion Sensor PS-2120
NOT INCLUDED, BUT REQUIRED:
1 850 Universal Interface UI-5000
1 PASCO Capstone UI-5400
Introduction:
The distances between the central maximum and the diffraction minima for a single slit are measured
by scanning the laser pattern with a Light Sensor and plotting light intensity versus distance. Also, the
distance between interference maxima for two or more slits is measured. These measurements are
compared to theoretical values. Differences and similarities between interference and diffraction
patterns are examined.
82
Theory
Single Slit Diffraction
When diffraction of light occurs as it passes through a slit, the angle to the minima (dark spot) in the
diffraction pattern is given by
a sin θ = m (m=1,2,3, …) Eq. (1)
where "a" is the slit width, θ is the angle from the center of the
pattern to a minimum, is the wavelength of the light, and m is
the order (m = 1 for the first minimum, 2 for the second minimum,
...counting from the center out).
In Figure 1, the laser light pattern is shown just below the
computer intensity versus position graph. The angle theta is
measured from the center of the single slit to the first minimum, so m equals one for the situation
shown in the diagram. Notice that the central spot in the interference Figure 1: Single-Slit
Diffraction pattern is twice as wide as the other spots since m=0 is not
a minimum.
Si m/L, where xm is the distance from the center of
central maximum to the mth minimum on either side of the central maximum and L is the distance
from the slit to the screen. Equation 1 now becomes
m = a sin θ = a tan θ = axm/L Eq. (2)
It is easier to measure the distance (2xm) from the mth minimum on one side to the m
th minimum on the
other side than to try to judge the center of the pattern. Equation 2 becomes:
m = a(2xm)/2L Eq. (3)
Our accuracy will be improved by making (2xm) as large as possible. The slit width is not known very
well. The uncertainty in the width is +/- 0.005 mm. That is a 25% uncertainty for the 0.020 mm slit. So
instead of using the slit width to calculate a value for the laser wavelength, we use the known
wavelength of the laser to calculate a more accurate value for the slit width. Rearranging Equation 3
yields:
83
Double-Slit Interference
When interference of light occurs as it passes through two slits, the angle from the central maximum
(bright spot) to the side maxima in the interference pattern is given by
d sin θ = n (n=0,1,2,3, …) Eq. (4)
where "d" is the slit separation, θ is the angle from the center of
the pattern to the nth maximum, is the wavelength of the light,
and n is the order (0 for the central maximum, 1 for the first side
maximum, 2 for the second side maximum ...counting from the
center out).
In Figure 2, the laser light pattern is shown just below the
computer intensity versus position graph. The angle theta is measured from the midway between the
double slit to the second side maximum, so n equals two for the situation shown in the diagram.
Figure 2: Double-Slit Interference
As before, theta is a small angle and Equation 4 may be rewritten:
= n/L Eq. (5)
where xn is the distance from the central maximum to the nth side maximum and L is the distance from
n+1 – xn - Eq. (6)
to the nth spot on the other side. Examination of Figure 2 shows that
And solving Equation 6 for the wavelength of the laser yields:
Eq. (7)
84
Note that the single slit diffraction pattern is also present in the double slit pattern. It is responsible for
the broad minimums that occur (see Figure 2). This means we must be careful when counting n in the
double slit pattern since a double slit maximum can be suppressed by a single slit minimum.
85
Figure 3: Setup
Setup
1. Mount the laser on the end of the optics bench. Mount the High Precision Single Slit disk to the
optics bench with the printed side toward the laser as shown. Turn on the laser. CAUTION: never
shine the laser beam directly into anybody’s eye! To select the desired slits, just rotate the disk until
it clicks into place with the 0.16 mm aperture slit illuminated by the laser.
Figure 4: Mounting the Light Sensor
2. Mount the Rotary Motion Sensor on the rack of the Linear Translator and mount the Linear
Translator to the end of the optics track (see Figure 4). Arrange things so the black stop block on
the linear translator arm is on the left side as viewed from the laser and all the way against the
bracket. Mount the Light Sensor with the Aperture Bracket (set on slit #1 = 0.1mm) in the Rotary
Motion Sensor rod clamp. The Light Sensor should be aligned with the bracket so it points parallel
to the optics track.
3. Move the light sensor until you can see the beam somewhere on the white screen. Use the
adjustment screws on the laser (see Figure 5) to adjust the position of the laser beam from left-to-
right and up-and-down to make the pattern on the white screen as bright as possible. Once this
position is set, it is not necessary to make any further adjustments of the laser beam when viewing
any of the slits on the disk. When you rotate the disk to a new slit, the laser beam will be already
aligned. Since the slits click into place, you can easily change from one slit to the next, even in the
dark.
86
Figure 5: Aligning the Laser
4. Move the Light Sensor up or down until the light pattern is centered on the slits shown in Figure 6.
Use slit #1 (=0.1 mm), not slit #4 as shown in Figure 6.
5. Set the Light Sensor for maximum sensitivity by pressing the 0-1 button. If the Light Intensity goes
too high (it will flat line at 100% on the graph), turn the sensitivity down by pressing the middle
button (0-100) on the Light sensor.
6. Plug the Rotary Motion Sensor and the Light Sensor into the PASPORT inputs on the 850 Universal
Interface.
Figure 6: Aligning
87
Conclusions:
1. Using your eyes, how does the single slit pattern change as you increase the slit size?
2. Using your eyes, how does the double slit pattern change as you increase the slit
separation?
3. On the Graph page, click on the Data Display button () to display multiple runs, then click
on the black triangle and select 20 Run. Click the Graph Re-scale button (). Now use the
black triangle of Data Display to select all four single slit runs. How does the Single Slit
Diffraction change as you vary the slit width (a)? Note that the 160 Run has a lower
Relative Intensity because we changes scales on the Light Sensor. Does this agree with
your answer to Question 1 above?
4. On the Graph page, use the Data Display black triangle to turn off the four single slit
patterns and turn on the 40 250 Run and the 40 500 Run. Click the Graph Re-scale button.
How does the Double Slit pattern change as you vary the slit separation (d)? Does this
agree with your observations from Question 2 above?
5. On the Graph page, use the Data Display black triangle to turn off the 40 250 Run and
turn on the 80 500 Run (40 500 still on). How does the Double Slit pattern change as you
vary the slit width (a)?
6. On the Graph page, use the Data Display black triangle to turn on the 40 250 Run and 40
Run (all others off). Can you see the single slit pattern in the double slit pattern? Why? If
the patterns do not exactly fall on top of each other, it is because you may not have started
the Rotary Motion Sensor at the same place each time, and the actual slit widths may not
be exactly the same.
7. On the Graph page, use the Data Display black triangle to turn on the 20 Run and the 40
Run (all others off). Click the Graph Re-
curve?
8. In the Legend box in the upper right click on the 20 Run to select it. Then click on the
Area Tool ( ). Now click on the 40 Run in the Legend box and click the area tool
again. The areas given are the area under each curve. Note that the area for the 40 Run is
roughly two times the area under the 20 Run. If the system where ideal, it would be
exactly two times. Why?
9. The manufacturer of the slits claims the uncertainty in the slit width is 0.005 mm. Do your
results agree with this? Compare the “a” column and the “slit width” column in the table
under the Data tab. What does this show about Equation (1)? What does it show about the
manufacturer?
88
10. The manufacturer of the slits claims the uncertainty in the slit separation is 0.01 mm. This
separation is 2%. In the
percentages since this is the largest uncertainly in the experiment. Do your values for the
wavelength agree with each other and with the known wavelength of the laser? What does
this show about Equation (4)? The uncertainty for the first 80 500 Run case (n~5) is
probably worse than stated since “right s” – “left x” ~ 0.0100 with an uncertainty of
89
90
Experiment 17: Radioactivity Simulation / Rolling Dice Experiment
References: Modern Physics, (Tipler & Llewellyn, 4th
ed.) Chapter 11, pp. 522–5377.
Introduction
The nuclei of many artificially created and of a few naturally occurring isotopes are unstable
and spontaneously change into other nuclei by the emission of α−particles, β−particles and
γ−ray photons in the process of radioactive decay. The rate at which decays occur, or number of
decays per unit time, is called the activity of the radioactive source. The half-life of a
radioactive substance is the time required for its activity to be reduced by one half. The intensity
of the radiation detected depends on the distance between the detector and the source (via the
inverse square law) as well as the activity of the source itself.
The probability of nuclear decay is determined by the interactions among the various
components within the nucleus, and because the forces among the nuclear par-ticles are so
strong, external forces do not usually play a role. Consequently, the prob-ability of decay is
independent of external circumstances, such as the number of other nuclei present, and the
number of nuclei of the original, or parent, isotope is a decreas-ing function of time. For
example, if 100 million nuclei were present in a sample, we might expect 100 of them to decay
in the next second. If only 50 million nuclei were pre-sent, we would then expect 50 to decay in
the next second. At each moment, the activ-ity of the source, A, is proportional to the number of
parent nuclei, N, remaining at that time:
where λ is called the decay constant. The negative sign indicates that the number of parent
nuclei decrease with time. Integration of equation (4.1) leads to the result that
where is the initial value of . Moreover, since the activity of the sample is propor-tional to N,
we can also write 0NN
where is the initial activity. The half-life, 0A21t, of a radioactive isotope can be derived from
equation (4.3) by setting A/A0
NOT: Because of the statistical nature of radioactive decay, there is an uncertainty inherent in
every count that is obtained. Repeated counts, under identical conditions, follow a Pois-son
distribution. Furthermore, the standard deviation of the Poisson distribution is equal to the
square root of the average number of counts. Consequently, the error in any sin-gle count is
taken as the square root of that count. Therefore, it is advantageous to maximize the number of
counts in order to reduce the error. To illustrate this, note that if 100 counts were obtained, the
uncertainty would be 10 counts — or 10%. If 10,000 counts were obtained, the uncertainty
would be 100 counts — only 1 per cent.
91
Units: An activity of one disintegration per second is called a becquerel (Bq). An older unit of
activity still in use is the curie (Ci), equal to 3.7 x 1010
disintegrations per second. A common
subunit is the microcurie: 1 μCi = 37,000 Bq and 1 Bq = 2.7 x 10-5
μCi.
M) tube connected to a pulse counter. A G–M tube is a cylindrical tube with a thin mem-brane
on one end containing a low-pressure gas. A wire through the central axis of the tube is held at a
high positive potential. When ionizing radiation enters the tube, the electrons freed through
ionization are attracted to the positively charged central wire. As they move toward the wire, the
electrons ionize other atoms through collisions, thus freeing more electrons which are also
attracted to the central wire and which in turn ion-ize other atoms. The resulting cascade of
electrons produces a small electrical pulse that is amplified and counted by a scalar–timer.
Objectives:
1.) Use non-radioactive materials (dice) to determine the probability that a single die will roll a
given number on a given toss in order to simulate when a single nuclei decays in a given period
of time.
2.) Demonstrate that the number, N, of dice remaining (number of nuclei not yet decayed) and
the rate of decay, dN/dt, both decrease exponentially.
3.) Determine the rate constant, k, and initial activity, Ao, for the dice (nuclei).
4.) Determine the experimental values for the half-life of the dice (nuclei) and calculate the
percent error for the experimental half-life values.
Background: While it is not possible to predict when any given radioactive nuclei will decay; it
is possible to predict with a high degree of probability the average rate of decay for a large
number of nuclei within a given sample. This is because the rate of decay for each type of
radioactive isotope is a constant and is a characteristic or that given isotope. The decay process
is then a statistical process. The decay of a radioisotope is a random event. That is, it is the
decay of a given nuclei is not dependent on the environment of the nucleus nor its past history.
One can then use statistical analysis to determine the probability of the rate of decay; likewise,
one can use statistical analysis to determine the probability that a given die will roll a specific
number when tossed. For dice, the probability that a specific number will be tossed is based
solely on the number of sides the die has. For 6-sided dice, there is a 1 in 6 probability (chance)
or rolling a 5 (or any of the numbers 1-6).
For example, if one took one hundred 6-sided dice and rolled them all at once and removed all
of the 5s that were face up, then how many of the 100 dice would one expect to remove? Since
the probability of rolling a 5 is 1 in 6, then multiply 1/6 by 100. The result is 16.6 or since dice
come in whole numbers about 16 or 17 dice would be removed. If 17 are removed from the
original 100, there would be 83 remaining. One could then predict in the next roll that the
number of dice that would land with the 5 face up would be 1/6 (83) or 13.8. Thus the rate of
decay is constant and can be used to simulate radioactive decay.
Experimental (Single Value)*: 1. Acquire 100 dice in a plastic bag, a plastic cup & data packet from the stockroom.
2. Pour all of the dice out of the bag into the cup.
3. Shake the dice in the cup.
4. Roll the dice out onto the lab bench. (Each roll simulates one minute.)
5. Remove all of the dice that landed with your given unknown number face up. Set them aside.
6. Record the number of remaining dice on your datasheet.
7. Put all of the remaining dice back into the cup.
8. Repeat steps 3-7 until all data is collected for first run.
9. For second (and third run), put all dice back into the cup and repeat steps 3-8.
10. Once data has been collected for three runs, average the numbers going across at each time.
Record the average in the proper column on the datasheet.
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11. Calculate the natural log of each value in the average counts/min column. Record in the
proper column on the datasheet.
12. Put dice back in plastic bag. Return plastic bag with dice and plastic cup to the stockroom.
*For the double and triple unknown numbers in step #5, simply remove all dice that apply to
the given unknown numbers.
T (number
of throws)
N (survived dice) Log N
0
1
2
3
4
5
6
7
8
9
10
11
12
Simulations
http://phet.colorado.edu/en/simulation/alpha-decay
http://visualsimulations.co.uk/software.php?program=radiationlab
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