physics and physical measurement - vienna...

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1 Physics and physical measurement

Range of magnitudes of quantities in our universe Physics seeks to explain the universe itself, from the very large to the very small. At the large end, the size of the visible universe is thought to be around 1025 m, and the age of the universe some 1018 s. The total mass of the universe is estimated to be 1050 kg.

The realm of physics1.1

Assessment statements1.1.1 State and compare quantities to the nearest order of magnitude.1.1.2 State the ranges of magnitude of distances, masses and times that

occur in the universe, from the smallest to the largest.1.1.3 State ratios of quantities as different orders of magnitude.1.1.4 Estimate approximate values of everyday quantities to one or two

significant figures and/or to the nearest order of magnitude.

How do we know all this is true?What if there is more than one universe?

A planet was recently discovered in the constellation Libra (about 20 light years from Earth) that has all the right conditions to support alien life. This artist’s impression shows us how it might look.

downloaded from www.pearsonbaccalaureate.com/diploma

UNCORRECTED PROOF COPY

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Physics and physical measurement 1

Rest mass is the mass of a particle when at rest; the mass increases if the particle moves fast enough.

If we can split an atom why can’t we split an electron?

The diameter of an atom is about 1010 m, and of a nucleus 1015 m. The smallest particles may be the quarks, probably less than 1018 m in size, but there is a much smaller fundamental unit of length, called the Planck length, which is around 1035 m.

There are good reasons for believing that this is a lower limit for length, and we accept the speed of light in a vacuum to be an upper limit for speed (3 108 ms1). This enables us to calculate an approximate theoretical lower limit for time:

time distance _______ speed

1035 m ________ 108 ms1 1043 s.

If the quarks are truly fundamental, then their mass would give us a lower limit. Quarks hide themselves inside protons and neutrons so it is not easy to measure them. Our best guess is that the mass of the lightest quark, called the up quark, is around 1030 kg, and this is also the approximate rest mass of the electron.

You need to be able to state ratios of quantities as differences of orders of magnitude. For example, the approximate ratio of the diameter of an atom to its nucleus is:

1010 m _______ 1015 m

105

105 is known as a difference of five orders of magnitude.

Some physicists think that there are still undiscovered particles whose size is around the Planck length.

What are the reasons for there being a lower limit for length?Why should there be a lower limit for time?

Production and decay of bottom quarks. There are six types of quarks called up, down, charm, strange, top and bottom.

Figure 1.1 The exact position of electrons in an atom is uncertain; we can only say where there is a high probability of finding them.



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This is not a small ratio; it means that if the atom were as big as a football pitch, then the nucleus would be about the size of a pea on the centre circle. This implies that most of the atoms of all matter consist of entirely empty space.

Another example is that the ratio of the rest mass of the proton to the rest mass of the electron is of the order:

1.67 1027 kg

_____________ 9.11 1031 kg

2 103

You should be able to do these estimations without using a calculator.

You also need to be able to estimate approximate values of everyday quantities to one or two significant figures.

For example, estimate the answers to the following:

How high is a two-storey house in metres?

What is the diameter of the pupil of your eye?

How many times does your heart beat in an hour when you are relaxed?

What is the weight of an apple in newtons?

What is the mass of the air in your bedroom?

What pressure do you exert on the ground when standing on one foot?

There is help with these estimates at the end of the chapter.

Measurement and uncertainties1.2

Assessment statements1.2.1 State the fundamental units in the SI system.1.2.2 Distinguish between fundamental and derived units and give

examples of derived units.1.2.3 Convert between different units of quantities.1.2.4 State units in the accepted SI format.1.2.5 State values in scientific notation and in multiples of units with

appropriate prefixes.1.2.6 Describe and give examples of random and systematic errors.1.2.7 Distinguish between precision and accuracy.1.2.8 Explain how the effects of random errors may be reduced.1.2.9 Calculate quantities and results of calculations to the appropriate

number of significant figures.1.2.10 State uncertainties as absolute, fractional and percentage uncertainties.1.2.11 Determine the uncertainties in results.1.2.12 Identify uncertainties as error bars in graphs.1.2.13 State random uncertainty as an uncertainty range () and represent it

graphically as an ‘error bar’.1.2.14 Determine the uncertainties in the slope and intercepts of a straight

line graph.

1 The diameter of a proton is of the order of magnitude ofA 1012 m. B 1015 m. C 1018 m D 1021 m.

Exercise

If most of the atom is empty space why does stuff feel so solid?



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The SI system of fundamental and derived units If you want to measure something, you have to use a unit. For example, it is useless to say that a person’s mass is 10, 60, 140 or 600 if we do not know whether it is measured in kilograms or some other unit such as stones or pounds. In the old days, units were rather random; your mass might be measured in stones, but your height would not be measured in sticks, but in feet.

Soon after the French Revolution, the International System of units was developed. They are called the SI units because SI stands for Système International.

There are seven base, or fundamental, SI units and they are listed in the table below.

Name Symbol Concept

metre or meter m length

kilogram kg mass

second s second

ampere A electric current

kelvin K temperature

mole mol amount of matter

candela cd intensity of light

Mechanics is the study of matter, motion, forces and energy. With combinations of the first three base units (metre, kilogram and second), we can develop all the other units of mechanics.

density mass _______ volume

kg m3

speed distance _______ time

m s1

As the concepts become more complex, we give them new units. The derived SI units you will need to know are as follows:

Name Symbol ConceptBroken down into base SI units

newton N force or weight kg m s2

joule J energy or work kg m2 s2

watt W power kg m2 s3

pascal Pa pressure kg m1 s2

hertz Hz frequency s1

coulomb C electric charge As

volt V potential difference kg m2 s3 A1

ohm resistance kg m2 s3 A2

tesla T magnetic field strength kg s2 A1

weber Wb magnetic flux kg m2 s2 A1

becquerel Bq radioactivity s1

Some people think the foot was based on, or defined by, the length of the foot of an English king, but it can be traced back to the ancient Egyptians.

The system of units we now call SI was originally developed on the orders of King Louis XVI of France. The unit for length was defined in terms of the distance from the equator to the pole; this distance was divided into 10 000 equal parts and these were called kilometres. The unit for mass was defined in terms of pure water at a certain temperature; one litre (or 1000 cm3) has a mass of exactly one kilogram. Put another way, 1 cm3 of water has a mass of exactly 1 gram. The units of time go back to the ancients, and the second was simply accepted as a fraction of a solar day. The base unit for electricity, the ampere, is defined in terms of the force between two current-carrying wires and the unit for temperature, the kelvin, comes from an earlier scale developed by a Swedish man called Celsius.



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Worked examples

1 Give units for the following expressed as (i) the derived unit (ii) base SI units:

(a) force

(b) kinetic energy.

2 Check if these equations work by substituting units into them.

(a) power work/time or energy/time

(b) power force velocity

Solutions1 (a) (i) N (ii) kg (m s2) or kg m s2

(b) (i) J (ii) kg (m s1)2 or kg m2 s2

2 (a) W : J/s or W : (kg m2 s2)/s or W : kg m2 s3

(b) W : N (m s1) or W : (kg m s2) (m s1) or W : kg m2 s3

In addition to the above, there are also a few important units that are not technically SI, including:

Name Symbol Concept

litre l volume

minute, hour, year, etc. min, h, y, etc. time

kilowatt-hour kWh energy

electronvolt eV energy

degrees celsius °C temperature

decibel dB loudness

unified atomic mass unit u mass of nucleon

Examiner’s hint: force mass acceleration.

Examiner’s hint: kinetic energy 1 _ 2 mv2

2 Which one of the following units is a unit of energy?A eV B W s1 C W m1 D N m s1

3 Which one of the following lists a derived unit and a fundamental unit?

A ampere second

B coulomb kilogram

C coulomb newton

D metre kilogram

Exercises



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Worked example

Convert these units to SI:.(a) year (b) °C (c) kWh (d) eV

Solution(a) 1 year 1 365 days 24 hours 60 minutes 60 seconds

3.15 107 s

(b) Here are some common conversions:

0 K 273 °C273 K 0 °C300 K 27 °C373 K 100 °C

(c) 1 kWh (energy) 1000 W (power) 3600 s (time)

3 600 000 J

3.6 106 J

(d) electrical energy electric charge potential difference

1 eV 1.6 1019 C 1 V

1.6 1019 J

The SI units can be modified by the use of prefixes such as milli as in millimetre (mm) and kilo as in kilometre (km). The number conversions on the prefixes are always the same; milli always means one thousandth or 103 and kilo always means one thousand or 103.

These are the most common SI prefixes:

Prefix Abbreviation Value

tera T 1012

giga G 109

mega M 106

kilo k 103

centi c 102

milli m 103

micro µ 106

nano n 109

pico p 1012

femto f 1015

Examiner’s hint: To change kilowatt-hours to joules involves using the equation:

energy power time.1 kW 1000 W and 1 hour 60 60 seconds.

Examiner’s hint: The electronvolt is defined as the energy gained by an electron accelerated through a potential difference of one volt. So the electronvolt is equal to the charge on an electron multiplied by one volt.

4 Change 2 360 000 J to scientific notation and to M J.

5 A popular radio station has a frequency of 1 090 000 Hz. Change this to scientific notation and to MHz.

6 The average wavelength of white light is 5.0 107 m. What would this be in nanometres?

7 The time taken for light to cross a room is about 1 108 seconds. Change this into microseconds.

Exercises

Examiner’s hint: The size of one degree Celsius is the same as one Kelvin the difference is where they start, or the zero point. The conversion involves adding or subtracting 273. Since absolute zero or 0 K is equal to 273 °C, temperature in °C temperature in K 273.



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Uncertainty and error in measurementEven when we try to measure things very accurately, it is never possible to be absolutely certain that the measurement is perfect.

The errors that occur in measurement can be divided into two types, random and systematic. If readings of a measurement are above and below the true value with equal probability, then the errors are random. Usually random errors are caused by the person making the measurement; for example, the error due to a person’s reaction time is a random error.

Systematic errors are due to the system or apparatus being used. Systematic errors can often be detected by repeating the measurement using a different method or different apparatus and comparing the results. A zero offset, an instrument not reading exactly zero at the beginning of the experiment, is an example of a systematic error. You will learn more about errors as you do your practical work in the laboratory.

Random errors can be reduced by repeating the measurement many times and taking the average, but this process will not affect systematic errors. When you write up your practical work you need to discuss the errors that have occurred in the experiment. For example: What difference did friction and air resistance make? How accurate were the measurements of length, mass and time? Were the errors random or systematic?

Another distinction in measuring things is between precision and accuracy. Imagine a game of darts where a person has three attempts to hit the bull’s-eye. If all three darts hit the double twenty, then it was a precise attempt, but not accurate. If the three darts are evenly spaced just outside and around the bull’s-eye, then the throw was accurate, but not precise enough. If the darts all miss the board entirely then the throw was neither precise nor accurate. Only if all three darts hit the bull’s-eye can the throws be described as both precise and accurate!

What conditions would be necessary to enable something to be measured with total accuracy?

Figure 1.2 All the players try to hit the bull’s eye with their three darts, but only the last result is both precise and accurate.

20512

914

11

816

719 3 17

2

1510

613

4

181 205

12

914

11

816

719 3 17

2

1510

613

4

181

20512

914

11

816

719 3 17

2

1510

613

4

181 205

12

914

11

816

719 3 17

2

1510

613

4

181

precise,not accurate

neither precisenor accurate

accurate,not precise

both accurateand precise



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It is the same with measurements; they can be precise, accurate, neither or both. If there have been a large number of measurements made of a particular quantity, we can show these four possibilities on graphs like this:

Significant figures

When measuring something, in addition to a unit, it is important to think about the number of significant figures or digits we are going to use.

For example, when measuring the width and length of a piece of A4 paper with a 30 cm ruler, what sort of results would be sensible?

Measurements (cm)Number of

significant figuresSensible?

21 30 1 2 yes

21.0 29.7 0.1 3 maybe

21.03 29.68 0.01 4 no

With a 30 cm ruler it is not possible to guarantee a measurement of 0.01 cm or 0.1 mm so these numbers are not significant.

This is what the above measurements of width would tell us:

Measurements (cm) Number ofsignificant figures

Value probably between (cm)

21 1 2 20–22

21.0 0.1 3 20.9–21.1

21.03 0.01 4 21.02–21.04

The number of significant figures in any answer or result should not be more than that of the least precise value that has been used in the calculation.

precise butnot accurate

true value ofmeasured quantity

number ofreadings

number ofreadings

number ofreadings

number ofreadings

accurate butnot precise

true value

neither accuratenor precise

true value

accurate and precise

true value

Figure 1.3 Here is another way of looking at the difference between precision and accuracy, showing the distribution of a large number of measurements of the same quantity around the correct value of the quantity.

If you are describing a person you have just met to your best friend, which is more important accuracy, precision or some other quality?



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Worked example

Calculate the area of a piece of A4 paper, dimensions 21 cm 29.7 cm. Give your answer to the appropriate number of significant figures.

Solution21 29.7 623.7

Area 620 cm2

6.2 102 cm2

Uncertainties in calculated resultsIf we use a stopwatch to measure the time taken for a ball to fall a short distance, there will inevitably be errors or uncertainties due to reaction time. For example, if the measured time is 1.0 s, then the uncertainty could reasonably be 0.1 s. Here the uncertainty, or plus or minus value, is called an absolute uncertainty. Absolute uncertainties have a magnitude, or size, and a unit as appropriate.

There are two other ways we could show this uncertainty, either as a fraction or as a percentage. As a fraction, an uncertainty of 0.1 s in 1.0 s would be 1 __ 10 and as a percentage it would be 10%.

These uncertainties increase if the measurements are combined in calculations or through equations. In an experiment to find the acceleration due to gravity, the errors measuring both time and distance would influence the final result.

If the measurements are to be combined by addition or subtraction, then the easiest way is to add absolute uncertainties. If the measurements are to be combined using multiplication, division or by using powers like x2, then the best method is to add percentage uncertainties. If there is a square root relationship, then the percentage uncertainty is halved.

Uncertainties in graphsWhen you hand in your lab reports, you must always show uncertainty values at the top of your data tables as a sensible value. On your graphs, these are represented as error bars. The error bars must be drawn so that their length on the scale of the graph is the same as the uncertainty in the data table. Error bars can be on either or both axes, depending on how accurate the measurements are. The best-fit line must pass through all the error bars. If it does not pass through a point, then that point is called an outlier and this should be discussed in the evaluation of the experiment.

Examiner’s hint: The least precise input value, 21 cm, only has 2 significant figures.

Examiner’s hint: Because we are using scientific notation, there is no doubt that we are giving the area to 2 significant figures.

8 When a voltage V of 12.2 V is applied to a DC motor, the current I in the motor is 0.20 A. Which one of the following is the output power VI of the motor given to the correct appropriate number of significant digits?

A. 2 W B. 2.4 W C. 2.40 W D. 2.44 W

Exercise



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Scalars are measurements that have size, or magnitude. A scalar almost always needs a unit. Vectors have magnitude and also have a direction. For example, a Boeing 747 can fly at a speed of 885 kmh1 or 246 ms1. This is the speed and is a scalar quantity. If the plane flies from London to New York at 246 ms1 then this is called its velocity and is a vector, because it tells us the direction. Clearly, flying from London to New York is not the same as flying from New York to London; the speed can be the same but the velocity is different. Direction can be crucially important.

1.0O

2.0time (s) �0.2

outlier

distance(m) �0.1

3.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

y

x

Motion showing a body travelling at a steady speedFigure 1.4 Error bars can be on the x-axis only, y-axis only or on both axes, as shown here.

Vectors and scalars1.3

Assessment statements1.3.1 Distinguish between vector and scalar quantities, and give examples of

each.1.3.2 Determine the sum or difference of two vectors by a graphical method.1.3.3 Resolve vectors into perpendicular components along chosen axes.



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Here is another example of the difference between a vector and a scalar. Suppose you walk three metres to the east and then four metres towards the north.

The distance you have travelled is seven metres but your displacement, the distance between where you started and where you ended up, is only five metres. Because displacement is a vector, we also need to say that the five metres had been moved in a certain direction north of east.

Here are some common examples:

Scalar Vector

Distance Displacement

Speed Velocity

Temperature Acceleration

Mass Weight

All types of energy All forces

Work Momentum

Pressure All field strengths

A vector is usually represented by a bold italicized symbol, for example F for force.

Free body diagrams

4 m north

distance walked � 7 mdisplacement � 5 m (north of east)

3 m east

5 m

Figure 1.5 Distance is a scalar, and in this case, the distance travelled is 3 m 4 m 7 m. Displacement is a vector, and here it is the hypotenuse of the triangle (5 m).

9 Which one of the following is a scalar quantity?A Pressure B ImpulseC Magnetic field strength D Weight

10 Which one of the following is a vector quantity?A Electric power B Electrical resistanceC Electric field D Electric potential difference

Exercises

weight � liftthrust � drag

lift

thrust of jets

weight

drag of air

weight � normal force

weight

normal orsupporting force

Figure 1.6 Free-body diagrams show all the forces acting on the body. The arrows should be drawn to represent both the size and direction of the forces and should always be labelled.(c) Aeroplane in level flight accelerating to the right:

(a) Book resting on a table: (b) Car travelling at constant velocity to the left:

weight � normal forcesdriving force � resistive forces

weight

normal forces

resistive forcesdriving force

of engine



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If two or more forces are acting at the same point in space, you need to be able to calculate the resultant, or total effective force, of the combination. The resultant is the single force that has the same effect as the combination.

If they are not parallel, the easiest way to determine the resultant is by the parallelogram law. This says that the resultant of two vectors acting at a point is given by the diagonal of the parallelogram they form.

You also need to be able to resolve, or split, vectors into components or parts. A component of a vector shows the effect in a particular direction. Usually we resolve vectors into an x-component and a y-component.

Worked example

A force of 20 N pulls a box on a bench at an angle of 60° to the horizontal. What is the magnitude of the force F parallel to the bench?

Figure 1.7 When the vectors are parallel, the resultant is found by simple addition or subtraction.

(a) (b)

(c)

2 N 3 N

resultant � 5 N to right

2 N 3 N

resultant � 1 N to left

3 N

3 N

6 N

resultant � zero

10 N

6 N resultant

magnitude of resultant � 14 N

60°

Figure 1.8 We can use a graphical method to find the resultant accurately.

Examiner’s hint: You can do this is by scale drawing using graph paper.

11 The diagram below shows a boat that is about to cross a river in a direction perpendicular to the bank at a speed of 0.8 ms1. The current flows at 0.6 ms1 in the direction shown.

The magnitude of the displacement of the boat 5 seconds after leaving the bank is

A 3 m. B 4 m. C 5 m. D 7 m.

Exercise

bank

bank

0.6 ms�10.8 ms�1

boat

y-component

x-component (F)A

B20 N

C60°

Figure 1.9 Resolving into components is the opposite process to adding vectors and finding the resultant.



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SolutionThe string will tend to pull the box along the bench but it will also tend to pull it upwards.

cosine 60° adjacent

__________ hypotenuse

F ___ 20

F 20 N cos 60° 10 N

Examiner’s hint: In the right-angled triangle ABC, the x-component (F) is adjacent to the 60° angle while the 20 N force is the hypotenuse.

12 A force of 35 N pulls a brick on a level surface at an angle of 40° to the horizontal. The frictional force opposing the motion is 6.8N. What is the resultant force F parallel to the bench?

Exercise Examiner’s hint: Here is an example of how not to answer a basic question:Find x.

x3 cm

4 cmHere it is

1 Which one of the following contains three fundamental units?

A Metre Kilogram Coulomb

B Second Ampere Newton

C Kilogram Ampere Kelvin

D Kelvin Coulomb Second

2 The resistive force F acting on a sphere of radius r moving at speed v through a liquid is given by

F cvr

where c is a constant. Which of the following is a correct unit for c?

A NB N s1

C N m2 s1

D N m2 s

3 Which of the following is not a unit of energy?

A W sB W s1

C k WhD k g m2 s2

4 The power P dissipated in a resistor R in which there is a current I is given by

P I 2R

The uncertainty in the value of the resistance is 10% and the uncertainty in the value of the current is 3%. The best estimate for the uncertainty of the power dissipated is

A 6%B 9%C 6%D 19%

Practice questions



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Here are some ideas to help you with the estimates on page 3:

1 How high is a two floor house in metres?

First we could think about how high a normal room is. When you stand up how far is your head from the ceiling? Most adults are between 1.5 m and 2.0 m tall, so the height of a room must be above 2.0 m and probably below 2.5 m. If we multiply by two and add in some more for the floors and the roof then a sensible value could be 7 or 8 m.

2 What is the diameter of the pupil of your eye?

This would change with the brightness of the light, but even if it were really dark it is unlikely to be above half a centimetre or 5.0 mm. In bright sunshine maybe it could go down to 1.0 mm so a good estimate would be between these two diameters.

3 How many times does your heart beat in an hour when you are relaxed?

You can easily measure your pulse in a minute. When you are relaxed it will most probably be between 60 and 80 beats per minute. To get a value for an hour we must multiply by 60, and this gives a number between 3600 and 4800. As an order of magnitude or ‘ball park figure’ this would be 103.

4 What is the weight of an apple in newtons?

Apples come in different sizes but if you buy a kilogram how many do you get? If the number is somewhere between 5 and 15 that would give an average mass for each apple of around 100g which translates to a weight of approximately 1 N.

5 What is the mass of the air in your bedroom?

To estimate this you need to know the approximate density of air, which is 1.3 kg m3. Then you need an estimate of the volume of your bedroom, for example 4 m 3 m 2.5 m, which would give 30 m3. Then mass density volume would give around 40 kg; maybe more than expected.

6 What pressure do you exert on the ground standing on one foot?

For this we would use the equation pressure force _____ area . The force would be

your weight; if your mass is 60 kg then your weight would be 600 N. If we take average values for the length and width of your foot as 30 cm and 10 cm, change them to 0.3 m and 0.1 m, and multiply, then the area is 0.03 m2. Dividing 600 N by 0.03 m2 gives an answer of 20 000 Pa.

You need to practise these kinds of estimations without a calculator.

If air is that heavy then why don’t we feel it?

How does the pressure exerted by one foot compare to blood pressure and atmospheric pressure?

What would happen to an astronaut in space if their space suit suddenly ripped open?



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