Mass in Motion

Malvern College; formerlyQ-Ievel Physics Project

d material for modernorswere closely associ ted

n ~hysics Project and thusof its spirit. These books are·

. al sense, nor do they givethat pupils will be carrying

they show the relevance andrid of the. principles studied

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LONGMAN PHYSICS TOPICS General Editor: John L. Lewis

[MASS IN MOTION [Jim JardineHead of the Physics DepartmentGeorge Watson's College, Edinburghformerly Scottish Team, Nuffield Physics Project

Illustrated by Geoffrey Salter

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LONGMAN GROUP LIMITEDLondonAssociated companies, branches and representatives throughout the world

© Longman Group Ltd 1970A II rights reserved. No part of this publicationmay be reproduced, stored in a retrieval systemor transmitted in any form or by any means -electronic, mechanical, photocopying, recordingor otherwise - without the prior permission ofthe copyright owner.

First published 1970

ISBN 0 582 32202 2

Printed in Great Britain byButler and Tanner Ltd, Frome and London

The author and publisher are grateful to the following for per-mission to reproduce copyright photographs: front cover FordMotor Company Limited; back cover Picturepoint Limited; page4 Teltron Limited; page 5 (left) Dr. Harold E. Edgerton,Massachusetts Institute of Technology; page 5 (right) StanleyRosenthal, Syndication International; page 6 ICI; page 7 (above)British Nylon Spinners Limited and G. Q. Parachute CompanyLimited; page 7 (below) Dunlop Company Limited; page 8 (above)British Hovercraft Corporation Limited; page 8 (below left)Associated Press Limited; pages 8 (below right) and 24 (above)Esso Petroleum Company Limited; page 9 National PhysicalLaboratory, Hovercraft Laboratory (Crown copyright reserved);page 10 UKAEA; page 13 (above left) Philip Harris Limited; page14 (left) Smiths Industries Limited; page 14 (right) John Emery,Glenalmond; page 14 (below) Venner Limited; page 15 PanaxEquipment Limited; page 17 Morris Laboratory InstrumentsLimited; page 18 (left) Strobe Automation Limited; page 19(above) BBC; pages 19 (below), 22, 23, 24 (below), 26 (below), 27,34 (right), 35, 36 (left), 44, 49,50 and 51 Heinemann EducationalBooks Limited, from Jardine Physics is Fun 1,2,3; pages 30, 33,36 (left), 38, 39, 40 and 41 Kodansha Limited, from Stroboscopeand Photographs of Physical Phenomena; pages 25 (left), 36 (right)and 37 Kodansha Limited, from Colour Slides of Physical Pheno-mena, distributed in the UK by Philip Harris Limited; page 26(above) British Leyland Corporation Limited; page45 USIS: page46 Kiekhaefer Mercury; Pages 48 and 58 (below) Science Journal;page 52 (above) Professor Lord Blackett FRS and the RoyalSociety; page 52 (below) CERN; pages 57 and 58 Road ResearchLaboratory, Crowthorne (Crown copyright reserved).

Weare particularly grateful to Heinemann Educational BooksLimited and to Kodansha Limited for their cooperation.

I ANSWERS I

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[,use the increase in speed is the same duringI of change of speed (acceleration) is constant.

rest the speed is directly proportional to thethe speed is not proportional to the distancever, proportional to the distance travelled

the images are equally spaced out.

Ilmselves have the same time intervals between

uld be taken of the white second hand of athe strobe lamp could be used to view a

tr second, or a flywheel rotating at a steadybe 'frozen' if the strobe lamp frequency were

two dimensions, this photograph shows thattie to m~ve at ~ constant speed in a straightlorce acting on It.

exerting a greater force which produces a

times the acceleration produced by one elastic

ird of that of the single trolley.

rduces acceleration of I ml s?If If If 3 ml s?

rt Fmls?

If f ml s?5 .

/I E ml s?m

n be attached by Sellotape to the end of the.sured. A graph of frequency against mass can

ached to the end of the blade and the vibrationbe found from the graph. You may like to try

lass (m) and also T' against m.

end on the Earth's gravitational pull. This

NOTETO THE

)TEACHER

I CONTENTS;

This book is one in the series of Physics background booksintended primarily for use with the Nuffield O-level PhysicsProject. Most of the team of writers who have contributedto the series were associated with that project. It wasalways intended that the Nuffield teachers' materialsshould be accompanied by background books for pupilsto read, and a number of such books are being producedunder the Foundation's auspices. This series is intendedas a supplement to the Nuffield materials - not booksgiving the answers to all the investigations pupils will bedoing in the laboratory, certainly not textbooks in the con-ventional sense, but books, easy to read and copiouslyillustrated, which show how the principles studied inschool are applied in the outside world.

The books are such that they can be used with con-ventional courses as well as with the new programmes.Whatever course the pupils are following, they often needstraightforward books to help clarify their knowledge,sometimes to help them catch up on any topic they missedin their school course. It is hoped that this series will meetthat need.

This background series will provide suitable materialfor reading in homework. This volume is divided intosections, and a teacher may feel that one section at a timeis suitable for each homework session.

ForcesMeasuring motionNewton's first and second lawsInertiaProjectilesExplosions and collisionsSummaryAnswers to questions in the text

414223034445961

FORCES To start a ball rolling you throw it or kick it. In each caseyou are exertingaforce on the ball. To make it changedirection you can head.the ball.

Again you are exerting a force on the ball. When youcatch a cricket ball you stop its motion by exerting aforce on it.

4

l ·__ - --"'-

\

In all the above cases forces are being used to changethe motion of a ball. Does the ball exert a force on you ineach case?' (NB: the answer to this and to the othernumbered questions in this book will befound on pp. 61-4.)

In the tube illustrated here, electrons are given off by ahot filament and speeded up by an electric force. The beamis then bent by a magnetic force produced by two largecoils.

Of course it is possible to exert a force on somethingwithout moving it. You can lean against a wall, squeeze arubber ball or twist a piece of plasticine, but even thenpart of the object moves with respect to the rest.

The next photograph shows a tennis ball which has beensquashed as it strikes a tennis racket. What is the importantdifference in the behaviour of rubber and of plasticine afterthey have been squashed? 2

IANSWERSI2

3

u

4

Slih:he

b.ay

5.a:e\st

6.\pi;

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{hen two bodies interact theother are equal in magnitude

~g the same law is to say thatin a collision'.

a(v - u)

tv - mu

l.hangeof momentum[he impulse.tgtn

in newtons/kilogramme is num-p:celeration of gravity g measured

and horizontal motion are in-I

mass m is moving with velocity v[his is called kinetic energy and its

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II

This photograph shows the titanium boom on theAmerica's Cup winner Intrepid bent under the action oftremendous forces. Titanium was chosen since it combinesstrength with flexibility.

5

table exerts an upward force on television set

6

When a body is stationary, either no forces act on it - anunlikely state of affairs - or the forces are balanced.

In the diagram left, two balanced forces keep thetelevision set at rest. If we consider the forces actingon the linesman's feet in the photograph as a single force,we can say that he is in equilibrium under the action ofthreebalanced forces. These forces are represented by threearrows in the diagram. Do you notice anything special aboutthe directions of these three forces?'

FRICTION

When a car runs out of petrol on a perfectly level road, iteventually stops. Its motion has been altered. A force, orforces, must have been acting on the car. What are some ofthese forces?'

The following pictures show how motion can be arrest-ed by solid to solid friction in a disc brake (left) or by airresistance in a parachute.

I SUMMARyl

t

high speed filmshow the motion.ts a concrete barrier at 60 km/h.~,singa seat belt! Explain why aie injury caused to a passenger

Something to doExamine the frictional forces between two flat pieces of wood, metal, glassetc. Can you find more than one way of reducing the friction?

If you can find an old dry wheel bearing (for example, in a bicycle or rollerskate), put a drop of oil on it and see the effect produced.

If forces are necessary to change the shape or motion ofa body, it might be interesting to see what happens if oneof these forces - friction - is reduced.

Something to doHere is a simple balloon puck you can build at home.Glue a cork in the centre of the rough side of a piece of hardboard, and thendrill a 3-mm hole through the centre of the cork and board. Fit an inflatedballoon on the cork so that the air escapes through the hole, and place thepuck on a smooth level surface such as a polished table. How does thepuck move when you give it a push?

I

I

J7

I FORCES I

8

The photograph shows a hovercraft moving on acushion of air. As air friction is very much less than thefriction between solid and solid, or solid and liquid, thehovercraft's driving engine will not need to exert a largeforce to keep it going.

A huge oil storage tank was recently floated on a cushionof air and then moved 350 metres by a small tractor.

The French Aerotrain is supported and guided by air-bearing pads, and is capable of speeds greater than300 km/h.

If it were possible to reduce the frictional forces completely,how much force would be needed to keep the Aerotrainmoving?' Can you give an example of a body moving withoutfrictioni"

EXPLOSIONSANDCOLLISIONS

total product Ft is often calledr, simply, the impulse..mentum will require a certainlained from a large force actinga small force acting for a long

I. cricket ball while keeping yourrill last for a short time and thelielarge and painful! If, however,rh the ball, the impulse will takeed will be smaller. The product~ch case, since it is equal to thelhe ball which finishes at rest in

from a wall to the ground, youID)rehitti~g the ground which ismlse. This change would happengs rigid, so that the force would

\~7ou usually bend your legs wheni; of the impulse is long and thefhe impulse would be the same;S straight or bent.rr to knock a nail into a plank ofentum of the hammer takes placelarge force is exerted on the nail.plank were resting on a piece of

r1as being hammered?"applies his brakes his car comesforce acts for a long time. In a

.h greater force acts for a shortlrnge of momentum (area underIe same.laboratory at Crowthorne, carsire concrete block to investigateand passengers. Dummies are

nd without safety belts.

I FORCES I FIELD FORCES

If we are going to think of a force as something whichchanges shape or motion, we will have to admit that someforces act through empty space. For example, you canpush a trolley without touching it, using two horse-shoemagnets as shown.

This photograph shows a vehicle propelled by a linearmotor which depends for its operation on powerfulelectromagnetic forces.

9

10

You have no doubt charged a plastic rod or pen andused it to repel or attract other rods or to pick up pieces ofpaper. The diagram left shows the dome of a Van deGraaff generator attracting soap bubbles. The muchbigger Van de Graaff generator in the photograph is usedby nuclear physicists to accelerate atomic particles. Ineach case electric forces are exerted.

- .- -

Very carefully designed experiments have shown thatthere is another force which always tends to draw all bitsof matter together-gravitational attraction. The experi-ment illustrated above can be used to measure this force.The heavier the bits of matter are, the greater is the attrac-tion; and the nearer they are together, the greater is theattraction. As this force is extremely small, a very finesuspension wire is essential.

The force between the adjacent spheres is measured bythe twist of the wire. A beam of light reflected from asmall mirror fixed to the suspension wire indicates theamount of twist.

If the force between two chunks of stone isF when they are 1 metre apart, it will beF"4 when they are 2 metres apart and

F"9 when they are 3 metres apart

What do you think the force would be if they were 4 metresapart?' This kind of change of force, with the square of thedistance, is called an 'inverse square law' relationship.


(

time

) _ mv - muI-

e change in momentum

Ict Ft is numerically equal to the

It stops accelerating and moveswill be no unbalanced force on

,ng machine show?" What will itand comes to rest ?43

Ijourney in the lift. Suppose it is!;celerates upward for 4 seconds,dy speed for 6 seconds, then

I and finally comes to rest. Sup-or 400 newtons. A graph of the

I(;!eighingmachine, measured inTin here.

3ady speedF-

rest

decelerati ngI I1 II 11 I1 I1 II I

U~ _.....J....._--'--_.....L..- ....•..1---'-_--J..._---'--8 10 12 14 16 18 20time (seconds)

I

ed force acting on you during ther' look like this.

time .(seconds)

FORCES

! 10 N

It is this force which holds you on the Earth's surface andcauses things to fall. As in the case of magnetic force andelectric force, this gravitational force acts through emptyspace. These three forces are sometimes calledfieldforces,and we refer to the regions in which the forces act asmagnetic fields, electric fields and gravitational fields.Gravitational force differs from the other two in that itis always a force of attraction and never of repulsion.

MEASURING FORCES

Science is concerned with measurement. Lord Kelvin oncesaid that unless you can measure something and expressthe result in numbers you have not advanced to the stageof science. To study forces, then, we must find some way ofmeasuring them.

Adding forcesWhen several forces (e.g. weights) act side by side, thecombined force is the sum of these forces.

w;: 130N

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11

I FORCES IIf one spring stretched by a certain amount supports

a weight W, then two identical springs stretched by thesame amount will support 2W, and so on.

12

Instead of springs we might use a number of identicalelastic bands each, say, 10 centimetres long. One wayof defining a unit of force might be to say that one unit offorce is needed to stretch one band until it was 15 centi-metres long. Here we are using the idea that forces changethe shape of a body.

F=3 units _. --t-E~=~~:~j-- F=3 units

If two such bands were placed side by side, two units offorce would then be needed to stretch both bands to 15centimetres. Three bands side by side would exert threeunits of force when stretched to 15 centimetres and so on.

A number of identical elastic bands could therefore beused to calibrate a spring in 'units of force'. In this waya simple spring balance could be constructed.


.;is a vector quantity. You muste pucks after the collision intorectionandat right angles to it. 40

r photograph of an atomic col-\d a stationary particle. What canl these part ides ?41

ration of momentum and mass/can interpret bubble chamber

t illustrated below. Many new-ered in this way.

- -- - ---- ----

Some examples of commercial spring balances, calib-rated in newtons, are shown in the photographs. It is im-portant to remember that spring balances measure forceeven if they are calibrated in mass units such as kilo-grammes.

About 1660 the British scientist Robert Hooke dis-covered that, when a spring was stretched, the increase inlength was related to the force in a simple way. Twice theforce produced twice the increase in length, three times theforce produced three times the increase in length, and soon. We could say that the increase in length of the springis directly proportional to the force applied. This statementis known as Hooke's Law.

There is, however, a limit to Hooke's Law. When do youthink it ceases to be true?"

Something to doSee how the strength of an elastic band varies with the force applied to it.Does it behave in the way described by Hooke's Law?

13

MEASURINGMOTION

14

L

The measurement of motion is much more difficult thanthe measurement of force, since it involves speed (dis-tance per unit time) and direction. Even if we restrictour present studies to the measurement of motion in astraight line, we still have to measure the distance travelledin a particular time interval. If it were always possible tofit a speedometer to the object we were studying, measur-ing speed would be simplified. This can be done with carsand even with expensive trolleys but it becomes rathermore difficult when dealing with bouncing balls or atomicparticles!

Fortunately there are several techniques which enableus to measure small time intervals fairly accurately, andfrom these speed can be deduced. Here are some of them.


s'l

vehicles of different masses and.s collide. They then move apart;:ermechanism enables us to take

1

0f the straws (attached to thethe collision.

eriment are shown in the aboveto measure the four speeds and

momentum before and after thefiomentum is a vector quantity and1m into account. 39

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STOP CLOCKSMEASURINGMOTION As hand-operated stop watches are not suitable for

measuring intervals ofless than a second, we use electrically-operated clocks such as the one illustrated at the foot ofp. 14 or the scaler shown below.

The scaler has a 1000 Hz oscillator which operates a set ofdials. If it runs for one second the dials read 1000, andso we can use this clock to read time intervals accuratelyto one thousandth of a second. If we operate any of thesedevices ourselves (e.g. by pressing a switch at the beginningand end of a certain interval of time) the result obtained isnot very accurate, as our own reactions are quite slow.The time between our seeing an event and responding toit by pressing the switch is called our reaction time.

Something to do.Devise an experiment to measure your reaction time, using a stop watch orother timing device.

15

MEASURINGMOTION

~I

16

Fortunately automatic methods of timing which do notinvolve human reaction time can be arranged. Thisdiagram shows a method of using the scaler as an elect-ronic clock to find the speed of a trolley. When the card,which is 10 centimetres long, interrupts a beam of light,the clock is switched on. The clock then runs until thecard passes out of the beam of light.

If the clock reads 50 milliseconds, the trolley hastravelled 10 centimetres in 50 milliseconds, which is 20centimetres in 100 milliseconds (O.ls) and therefore 200centimetres in 1 second.

Its average speed is therefore 2 metres/ second. Whydo we say average speed?"


lJJI

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I

caused by a large mass movingss moving very quickly. Noticetainless steel plate exposed out-8) orbiting 400 kilometres abovewas caused by a 10-7 g micro-out 20 km/s.ige caused by a car crashing intor./h, engineers at Cornell Aero-.:ieda vehicle on to a horizontallyheight must the car be droppedfor

The diagram at the foot of p. 16 shows the scaler beingused as an electronic clock to find the speed of a rifle bullet.As it shoots through a thin aluminium foil, the bulletbreaks one circuit and starts the clock. After travelling onemetre it breaks another circuit, in a similar way, and stopsthe clock.

MEASURINGMOTION

*

TICKER TAPE

When a ticker timer is wired to a 50 Hz supply the armvibrates up and down fifty times a second. This vibratingarm is used to mark a paper tape every fiftieth of a secondas it passes through the timer. By measuring the separationof the dots on the paper tape, the distance it has travelledevery fiftieth of a second can be found. It is oftenconvenient to find the distance gone in 10 fiftieths of asecond (a 'tentick') and to express the speed in centimetresper tentick.

0 2 3 4 5 6 7 8 9 10 11 J

i . ~t ~

1 tentick I-1

17

~" " ~

tape A

tape B

18

We measure the length of 10 gaps between, say, tickso and 10 or 1 and 11 and not the distance between ticks1 and 10. What is the time interval between ticks 1 and10?10

The distance between the dots depends on the speed atwhich the tape is moving. Which tape in the diagrams on theleft was moving at the greater speed?'!

MULTIFLASH PHOTOGRAPHY

The stroboscopic or multiflash photograph providesone of the most flexible methods of studying motion. Alamp which flashes at regular time intervals (left) is used toilluminate a moving object, and a time-exposure photo-graph is taken. Alternatively, a camera (below) with a

rotating disc in front of its open shutter can be used tophotograph a fully lit moving object. In each case a seriesof photographs is taken at regular time intelvals on thesame negative. When the images are close together the ob-ject is moving slowly, and when they are far apart it is mov-ing quickly. Study the next photograph, which was taken inthis way, and see ifyou can tell when the tennis racket ismov-ing slowly, when it is speeding up and when it is slowingdown. What can you say about the movement of the ball?'?

If we know the time interval between the images, andfrom the scale of the photograph can find the distancetravelled, we can calculate the average speed during eachtime interval.

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EXPLOSIONSAND,COLLISIONS

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pal

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.ugh the water as the propellerposite direction.

iooth level surface and throw some heavy, happens. Why do you not normally observeeven a much heavier object?stopper of a plastic bottle, a simple 'rocket'

ith water and pump in air until the cork isxperiment out of doors!it into the air with its mouth open. Explain

This toy car was photographed every tenth of a second.as it moved past a half-metre stick. What, very roughly,was its average speed?" Was it going at a steady speed allthe time, or was it speeding up or slowing down?"

19

time

v=o

ACCELERATION

Imagine a car starting from rest on a level road. A camerais set to take a photograph of the speedometer every twoseconds. The above diagrams show the results that, incertain circumstances, might be obtained.

A graph showing how the speed varies with time is shown,left. Describe this motion. 15 Why has the graph this shaper!"

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time (seconds)

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t=3s

3

A trolley fitted with a speedometer is allowed to rundown an incline and the speed is noted every second. Agraph showing how the speed varies with time is shownleft. Again we have constant acceleration, this timebecause of the gravitational force acting on the trolley.This agrees with the relation v = u + at, when u = O.


part, the product of the mass (m)is found to be the same.

losions' this product mv is foundn a special name: momentum.o account, we have

- m'v'- m' a' t

s start from rest and that a is thetogether for a short time t

= -m'a'

ond law (F = m a) shows us that

= -F'

trolleys at any instant are there-csite in direction. This is really athird law of motion: 'to everynd opposite reaction'.a small carbon dioxide cylinder.

of the cylinder in one direction,rhe trolley, are propelled in the-lley accelerates, momentum m ve m a acting on the trolley at anyn size and opposite in directioncarbon dioxide.

V=·~O

If the speed has been noted every metre as in thediagram above, we might have had

speed after travelling 1 metre = 1 m/ sspeed after travelling 2 metres = 1.4 m/sspeed after travelling 3 metres = 1.7 m/s

2·0 3·0

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S EN~

"0 "0Q) Q)Q) Q)

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o 32 3 o 2

distance (metres) distance (metres)

The left-hand graph shows the average speed plottedagainst distance, and the right-hand graph shows thesquare of the speed plotted against distance. Which graphshows direct proportionalityr'? This agrees with the relationv2 = u2 + 2as, when u = O.

21

_____J

Throw a handful of coins into the air. Watch them rising,spinning, falling, rolling, sliding ... stopping. There aremany forces acting on the coins, but the coins soon cometo rest. Why? All our everyday experiences lead us to thesame conclusion that, left to their own devices, things willeventually stop moving. It is little wonder, then, that forcenturies it was thought that a force was needed to keepthings moving. Today, of course, we know that a spaceship will travel from the Earth to another planet and that,once it is a reasonable distance from the Earth, it will movefreely without any force being needed to push it. As theEarth's gravitational field is acting on everything on theEarth's surface, we can never observe the motion of abody on which no forces are acting. So we do the next bestthing and try to balance the forces acting on the body.

NEWTON'SFIRST ANDSECONDLAWS

In the linear air track, a light plastic vehicle is supportedon a cushion of air in such a way that the weight of thevehicle is exactly balanced by the force of the air pushingup on the vehicle. If the force of the air is increased, thevehicle rises until the two forces are again balanced. Ifthis air force is reduced, the vehicle falls very slightly sothat the upthrust increases until the two forces againbalance.

22

PROJECTILES

t is observed if the gun fires the.hotographs on pages 40-1 showd regardless of the speed at which

led the 'monkey and hunter' ex-monkey is hanging from a branch

hunter is about to shoot him.~.;esthe flash of the gun he dropsuonkey and the bullet will bothsnce in the same time, so that theII. the hunter will not succeed. Ofgun must not have been adjustedvity into account!

derstand this more clearly if you!ation taking place in a giant lift.the very moment the gun is fired, then the lift and all it containsrd at approximately 10 metres/ere in the lift, you would see the»ss the lift and strike the monkey,,~ing' beside the branch of the tree.fr outside the lift could watch what, of course, see the monkey andthe only difference being that thee falling too!

-- ---~'\

I

NEWTON'S FIRSTAND SECOND LAWS weight of vehicle

force of air

Air jets along the sides of the track produce balancedsideways forces in a similar way, so that the vehicle issupported on a cushion of air and is free to move in onlyone direction - along the length of the tube. If we ignorethe slight effects caused by air resistance, we can say thatno forces act on the vehicle along the length of the tube.

This is a multiflash photograph of a straw attached toan air track vehicle. What can you say about the motion ofthe vehicle?" What is your reason for saying thisi'? Whatassumption have you made about the flasherr'" How wouldyou check that it is valid?"

The air track vehicle is free to move in only one dimen-sion. If, however, we float a ring magnet on a cushion ofcarbon dioxide gas, it is free to move in two dimensionsover the surface of a sheet of glass.

solid CO2

23

NEWTON'S FIRSTAND SECOND LAWS

24

If the apparatus illustrated above is used to photo-graph a single puck moving on the plate of glass, the resultis as shown in the next photograph. What additional in-

formation does this photograph give about the motion of abody when no unbalanced force acts on it?22

The next multi-flash photograph shows the motion of aball-bearing when the gravitational force acting on it isexactly balanced by fluid friction. The constant speedproduced is called the terminal velocity. After falling forsome time a raindrop or a parachute will reach a terminalvelocity because of air resistance.

'e have seen, is no respecter ofboth ball-bearings, so that afterve fallen through the same dis-.ed in the photograph.ectromagnet was used to releaseat the instant the other left the

Although it is not possible in a school laboratory to getrid of the effects of gravity, experiments conducted inspace-craft and free-fall laboratories support the beliefthat, when a body is completely free to move in threedimensions, it will travel. at a constant speed in a straightline.

Here you can see a day's ration of freeze-dried foodpellets in mid-air during a zero-g flight.

Three hundred years before space travel became areality, Isaac Newton was able to say that 'a body will stayat rest or move with a constant speed. in a straight lineunless an unbalanced force acts on it'. This statement isNewton's first law of motion.

25

ACCELERATING A BODY

Here are three cars of similar mass, each with a differentengine. All are capable of travelling at 110 km/h, which is'the maximum permitted speed on Britishroads. Why th\n dothe manufacturers fit a 76 bhp engine to the Cooper S modelwhen this speed can be reached with a 38 bhp engine in thestandard Mini?"

Acceleration and forceTrolley experiments show that if one unit of force (1 elasticthread) produces a certain acceleration, two units of force(2 elastic threads) produce twice the acceleration (see thediagrams). Of course other factors such as mass mustremain the same. If three elastic threads, rather than one,were used to pull the trolley, how much greater would theacceleration be?"

1 elastic thread

cc.g(f) co.~~:;l Q)

U("\jU

co

2 elastic threads

26

I PROJECTILES I

.,

Jt

t tt t

f t, t

f ,t t

correct, the speed at which therorizontally should not affect theis is confirmed above, where a:ontal velocities was used.


1 trolley

This result could be expressed by saying thatthe accelera-tion of a body is directly proportional to the net (or un-balanced) force acting on it. In other words, double theforce: double the acceleration, and so on.

cc.g'" ro

.~~::;J Q)

UNU ro

2 trolleys

Acceleration and massUsing the same apparatus as before, you find that twoelastic threads produce two units of acceleration whenthey pull one trolley. If, however, the same force is exertedon two trolleys, only one unit of acceleration is producedDoubling the mass has halved the acceleration.

27


28

If the same two threads were used to pull three trolleys, howwould the acceleration produced compare with the accelera-tion of one trolley?"

This result could be expressed by saying that theacceleration produced by a constant net (or unbalanced)force is inversely proportional to the mass being accelera-ted. In other words, double the mass: half the acceleration,and so on.

The results of the two trolley experiments could be sum:marised by saying that the acceleration a of a body isdirectly proportional to the net force applied F, and isinversely proportional to the mass m.

" . FI.e. a IS proportional to -

mThis is really a statement of Newton's second law ofmotion.

THE NEWTON

The unit of mass is the kilogramme. This is defined as themass of a platinum block kept at the International Bureauof Standards at Sevres near Paris.

The metre is the unit of length and is defined in terms ofa particular wavelength in the spectrum of krypton.

The second is based on the natural frequency of thecaesium atom.

From these three basic SI units, we can now define aunit of force. It is called appropriately after Sir IsaacNewton. When an unbalanced (resultant or net) force of1 newton acts on a mass of 1 kilogramme, the accelerationproduced is 1 metre per second per second.

Use the relationship a is proportional to .!. to completethe following table." m

1 N acting on 1 kg produces acceleration of 1 m/ S2

3 N " " 1 kg" " " rn/s-FN " " 1 kg" " " m/s-FN " "5 kg" " " rn/s-F N " "m kg" " " m/ S2

I PROJECTILES Ifl.

n

trU

fiab

lis time the camera was placed in.'2eball rolled toward the camera.t. You will see that this pictureleft-hand photograph on p. 34.vertical motion, in this case thes'ravity, is not affected by thelievertical and horizontal move-

of apparatus was used whichhorizontally and, at the sametationary ball-bearing. A multi-


-- --_ .. " ------.~'\

From the last line of this table you can see that___(newtons)

~(metres/ second" )----

or F =ma ~(kilOgrammes)

Other systems of units may be used with this relation-ship, but we shall use only SI units in this book.

One of the best known of all the stories about Newtontells of the time when he was forced to leave Cambridgebecause of the plague and return to his home in Lincoln-shire. As he watched an apple fall from a tree one day hewondered if the same force might not keep the Mooncircling the Earth. Whether or not such an incident reallymarked the beginning of Newton's theory of gravitationis not very important, but the story might help you toremember that 1 newton is approximately the forceexerted by gravity on an average-sized apple. That is, theweight of an apple is about 1 newton.

29

INERTIA

30

In the previous section we discussed how an unbalancedforce applied to a body caused it to accelerate. The amountof acceleration depended on the mass of the body: abody of small mass was easy to accelerate; a body oflargemass was much more difficult. This property of a body toresist acceleration - or, of course, deceleration - is oftencalled its inertia. Scientists sometimes refer to the inertialmass of a body.

You may have tried pushing various masses on a traywhich was covered with ball bearings to reduce friction.Applying a small force to each of these masses enabled youto feel how their inertias differed, that is, how reluctant orotherwise they were to change their motion. The largermasses were much more difficult to accelerate. We mightsay: a body of low inertia (i.e. small mass) is easy to ac-celerate: a body of high inertia (i.e. large mass) is dif-ficult to accelerate.

When one block in a pile of blocks is given a sharpknock, those above it do not move with it although thefrictional forces acting between the blocks are muchgreater than the air resistance. The top four blocks are'reluctant' to move from their original position. That is,they have inertia.

\ PROJECTILES I

Oi

grrespe;n:

tiredi:

shabIS

th:

a ball is dropped it acceleratesoultiflash photograph (left) showsa ball beside a metre stick. Studyind then decide which of the follow-'all are correct. 32

teadily.ant.constant.

each time interval increases by a

ring each time interval increases

tal to the time it has been falling.val to the distance it has fallen.:!lto the square root of the distance

try to calculate the acceleration ofh. To do this you will need to knowimages. It is 0.033 seconds. 33

saw that the mass of a body doesill in a gravitational field. Therbled on twice the mass, trebledo on. If we ignore air resistance,t this section, we can say that allir size or mass, accelerate towardteo

Something to doTry to remove the card shown here so that the coin falls straight down intothe tumbler.

Mass is sometimes defined as 'the amount of matter in abody'. This is not a very useful definition as it does not tellus how to measure the mass. If we could easily count the

, number of nucleons (that is, the protons and neutrons) ina substance, we might use this as a measure of mass.However, we cannot do this. Fortunately the 'amount ofmatter' in a body affects the ease with which it can beaccelerated or decelerated, and it is this property ofmatter - its inertia - which we use to measure mass. Forthis reason the terms 'inertia' and 'mass' are often inter-changeable.

INERTIAL BALANCE

If you have a device for measuring force F, such as a springbalance, and a method of calculating acceleration a,perhaps a ticker timer or strobe photogrph, then you canuse the relationship F = ma or m = 7i to measure themass or inertia of the body.

Something to do~ Here is a simple experiment you might like to try.

Clamp a hacksaw blade to a table leg and fix a lump of plasticine to theend of it. Find how many to and fro swings there are each second (i.e. thefrequency). If you put a larger lump of plasticine on the end, would you expectthere to be more or less resistance to the change of speed during each to and fromovement? Would you expect this to increase or decrease the frequency? Tryit and see if your prediction is correct.

In the project above you have built a simple inertialbalance. Can you think how it might be calibrated to enableyou to measure unknown masses?" Would it work just aswell on the moon" or in a space ship in outer spacei/? Doesthe operation of this balance depend at all on the pull due togravity, or is it independent of it ?30

31

\ INERTIA I

Something to doThis diagram shows an alternative form of inertial balance. Long elastic threadsor springs are attached to a trolley or toy car which is then loaded so that thetotal mass is increased. How does the loading affect the to and fro frequency?How could this be refined to measure mass?

GRAVITATIONAL FIELD

When a mass is in a gravitational field, there is a forceacting on it. This gravitational force is something we arevery familiar with and we call it the weight of the body.

The mass of a body is the same everywhere, but theweight of the body will depend on the strength of thegravitational field. As the Moon's gravitational fieldhas a different strength from that of the Earth, the weightof a body on the Moon will be different from its weight onthe Earth.

A multiflash photograph of two spheres, one bigger andheavier than the other, is shown opposite. The accelerationis clearly the same for the small mass m and the large massM. What does this tell you about the size of the force acting on

- force feach sphere ?31 But acceleration = -- = - for one bodymass m

For M for the other body. As they have the same accelera-

tion, then force/ mass -has the same value for differentbodies in a gravitational field. We use this as a way ofmeasuring the strength of a gravitational field. In futurewe will measure field strength in newtons per kilogramme.

32

I~ - -- ----

(n

isti.n-grfiroc

Ns'

giTc-

Inace

BL

arthth

t of

,~ gr.gr

form of inertial balance. Long elastic threadsor toy car which is then loaded so that the

the loading affect the to and fro frequency?ire mass?

IELD

svitational field, there is a forcetional force is something we areve call it the weight of the body.

l

is the same everywhere, but thedepend on the strength of thethe Moon's gravitational fieldrom that of the Earth, the weightill be different from its weight on

rh of two spheres, one bigger andl0hownopposite. The accelerationsmall mass m and the large mass

about the size of the force acting on. force f

rauon = -- = - for one bodymass m

. As they have the same accelera-

las the same value' for differentI field. We use this as a way ofcf a gravitational field. In futureength in newtons per kilogramme.

Using g for the gravitational field strength, we have(newtons/kilogramme) (newtons)

(kilogrammes)

Thus, at the surface of the Earth where the field strengthis about 9.8 newtons/kilogramme, there will be a gravitational force of 9.8 newtons on 1 kilogramme, 2 X 9.8newtons on 2 kilogrammes, 3 X 9.8 newtons on 3 kilo-grammes and so on. Since on the Moon the gravitationalfield is about one sixth of its value on the Earth, the forceon each mass will be only one sixth as great.

Numerical equivalence of gravitational fieldstrength and acceleration of a falling body

Suppose a body of mass m kilogrammes is put in agravitational fieldwhose strength isg newtons/ kilogramme.The force caused by gravity will be m X g newtons. This iscalled the weight of the body.

If a body is released and allowed to fall freely under theinfluence of this force, the body will accelerate. Thisacceleration is given by

Fa =-

m

But as !... is also the gravitational field strength g, them

acceleration of gravity must be g metres/second". That is,the acceleration in m/ S2 has the same numerical value asthe gravitational field strength in N/kg. Thus at the surfaceof the earth, where the field strength is 9.8 newtons/kilo-gramme, the acceleration of a body falling freely undergravity is 9.8 metres/second".

33

J

I PROJECTILESI

34

You have seen that when a ball is dropped it acceleratestowards the ground. The multiflash photograph (left) showsa series of images of such a ball beside a metre stick. Studythe photograph carefully, and then decide which of the follow-ing statements about the ball are correct. 32

1. Its speed is increasing steadily.2. Its acceleration is constant.3. The force acting on it is constant.4. The distance it drops each time interval increases by aconstant amount.5. Its average speed during each time interval increasesby a constant amount.6. Its speed is proportional to the time it has been falling.7. Its speed is proportional to the distance it has fallen.8. Its speed is proportional to the square root of the distanceit has fallen.

You might also like to try to calculate the acceleration ofgravity from the photograph. To do this you will need to knowthe time interval between images. It is 0.033 seconds. 33

In the last chapter we saw that the mass of a body doesnot alter its acceleration in a gravitational field. Thegravitational force is doubled on twice the mass, trebledon thrice the mass and so on. If we ignore air resistance,as we will do throughout this section, we can say that allbodies, regardless of their size or mass, accelerate towardthe Earth at the same rate.

- ----~-

a

I!I

HC

1b,P'u

5,

'- Hoe[[~

fr,thfir

it

t,

b:l

YOtW

g'

discussed how an unbalancedsed it to accelerate. The amounton the mass of the body: a

sy to accelerate; a body oflargecult. This property of a body tocourse, deceleration - is often

s sometimes refer to the inertial

rshing various masses on a traycall bearings to reduce friction.ach of these masses enabled youiiffered, that is, how reluctant orIUlangetheir motion. The largerIifficult to accelerate. We might(i.e. small mass) is easy to ac-inertia (i.e. large mass) is dif-

pile of blocks is given a sharpnot move with it although the

between the blocks are muchstance. The top four blocks aretheir original position. That is,

I PROJECTILES I Now let us consider what happens when a ball is pro-jected horizontally and then allowed to fall. For example,we might roll a ball along a table and then allow it to runoff the end. The picture at the foot of p. 34 shows a multi-flash photograph of such a ball. It is taken from the sideof the table.

By drawing equally spaced vertical lines on the photo-graph, you can see that the horizontal speed of the ballremains constant. That is, it continues to move at the samespeed in its original direction. This is what you might haveexpected from Newton's first law of motion, since there isno horizontal unbalanced force acting on the ball.

So it would appear that the speed of a body in one direc-tion is not affected by a force acting at right angles to thatdirection.

This is confirmed by the photograph of the same eventshown below. To take this picture the camera was heldabove the table, so that the horizontal velocity of the ballis shown before and after leaving the table. You can seethat it is constant throughout.

To investigate the vertical motion of the ball a third

35

~ ~-~~ - --- - -- -----_._-----------

36

1--':

photograph was taken. This time the camera was placed infront of the table so that the ball rolled toward the camera.

The result is shown left. You will see that this picturelooks very similar to the left-hand photograph on p. 34.Does this mean that the vertical motion, in this case theacceleration caused by gravity, is not affected by thehorizontal motion? Are the vertical and horizontal move-ments quite independent?

To check this, a piece of apparatus was used whichprojected a ball-bearing horizontally and, at the sametime, released a second stationary ball-bearing. A multi-


tbsJ

c

ISre:;

ere used to pull three trolleys, howiuced compare with the accelera-

expressed by saying that thea constant net (or unbalanced)nal to the mass being accelera-the mass: half the acceleration,

olley experiments could be sum:he acceleration a of a body is.he net force applied F, and isihe mass m.oportional to F

mnt of Newton's second law of

ilogramme. This is defined as thekept at the International Bureauar Paris.length and is defined in terms ofthe spectrum of krypton.

.n the natural frequency of the

c SI units, we can now define ad appropriately after Sir Isaacanced (resultant or net) force ofof 1kilogramme, the accelerationsecond per second.

is proportional to .! to completem

-es acceleration of 1 m/ S2

" " m/s"" " m/s?" m/s-

m/s-"

" "

I PROJECTILES Iflash photograph of the event is shown here. You cansee by comparing the heights of the ball-bearings atdifferent times that they fall with the same verticalacceleration. So the vertical and horizontal motions areindependent.

The grid superimposed on the same photograph makesthis point clear. The horizontal speed is constant, as nounbalanced force is acting in that direction (Newton'sfirst law), and the vertical acceleration is constant becausea constant vertical force (gravity) is acting on the ball-bearing (Newton's second law).

37

~ROJECTILES I

If these statements are correct, the speed at which theball-bearing is projected horizontally should not affect thevertical acceleration. This is confirmed above, where anumber of different horizontal velocities was used.

38


c.g'" ro

.~~:;:J Q)

UNU ro

1 trolley

t

BODY.milar mass, each with a differentif travelling at 110 km/h, which is"peed on British roads. Whythen dobhp engine to the Cooper Smodel

rached with a 38 bhp engine in the

rcew that if one unit of force (1 elasticlin acceleration, two units of forceice twice the acceleration (see thether factors such as mass mustze elastic threads, rather than one,olley, how much greater would the

[ PROJECTILES [ A multiflash photograph of a ball-bearing fired into theair can be seen here. Which of the following statements aboutthe ball-bearing are correct ?34

1. Its deceleration as it rises is numerically the same as itsacceleration as it falls.2. Its vertical motion is independent of its horizontal motion.3. It is speeding up as it falls.4. Its horizontal speed is constant.5. Its vertical speed is constant.6. At any particular height above the point of projection,its speed is the same when it is rising as it is when it isfalling.

Suppose that in the experiment illustrated on p. 37 thestationary ball-bearing is released from another pointwhich is in the direct line of fire of the projected ball. Then,if there were no gravitational field, the projected ball-bearing would strike the stationary one.

'.

1111BlIi DII1I.' ,'Il• ,,

.'.01

t

•III'! %'I

I!': iii 1!':1II

••Iii

"""'"

39

//

I PROJECTILES I

Gravity, however, as we have seen, is no respecter ofpersons. It acts equally on both ball-bearings, so that aftera given time both will have fallen through the same dis-tance. The result is indicated in the photograph.

In this experiment an electromagnet was used to releasethe stationary ball-bearing at the instant the other left themuzzle.

40

rst'

pe

resatunNe

rated above is used to photo-g on the plate of glass, the resulthotograph. What additional in-

srapli give about the motion of aorce acts on it?22'otograph shows the motion of aavitational force acting on it isd friction. The constant speedrminal velocity. After falling forparachute will reach a terminal.istance.

---- --- -------

41

I PROJECTILES IExactly the same result is observed if the gun fires the

'bullet' at an angle. The photographs on pages 40-1 showthat a direct hit is obtained regardless of the speed at whichthe bullet is fired.

This is sometimes called the 'monkey and hunter' ex-periment. Imagine that a monkey is hanging from a branchof a tree and sees that a hunter is about to shoot him.Whenever the monkey sees the flash of the gun he dropsfrom the branch. The monkey and the bullet will bothhave fallen the same distance in the same time, so that themonkey's attempt to foil the hunter will not succeed. Ofcourse, the sights of the gun must not have been adjustedto take the effects of gravity into account!

You can probably understand this more clearly if youimagine the whole operation taking place in a giant lift.If the lift rope breaks at the very moment the gun is firedand the monkey lets go, then the lift and all it containswill accelerate downward at approximately 10 metres/second/second. If you were in the lift, you would see thebullet move straight across the lift and strike the monkey,which would be still 'hanging' beside the branch of the tree.

If a stationary observer outside the lift could watch whatwas going on he would, of course, see the monkey andbullet falling as before, the only difference being that thehunter and tree would be falling too!

42


into the air. Watch them rising,sliding ... stopping. There arecoins, but the coins soon come

Lydayexperiences lead us to thero their own devices, things willis little wonder, then, that for

;iat a force was needed to keepcourse, we know that a space

arth to another planet and that,.nce from the Earth, it will moveiieing needed to push it. As theis acting on everything on the

never observe the motion of aire acting. So we do the next bestthe forces acting on the body.

light plastic vehicle is supported.ch a way that the weight of the.d by the force of the air pushingforce of the air is increased, theo forces are again balanced. Ifthe vehicle falls very slightly so.ses until the two forces again

PROJECTILES

I j

"

43


When two trolleys spring apart, the product of the mass (m)and the speed (v) of each is found to be the same.

44

Because in all similar 'explosions' this product mv is foundto be conserved, it is given a special name: momentum.

If we take direction into account, we have

mvor m at

- m'v'- m' a' t

Assuming that the trolleys start from rest and that a is theacceleration as they react together for a short time t

m a = =m' a'

which from Newton's second law (F = m a) shows us that

F = -F'

The forces acting on the trolleys at any instant are there-fore equal in size but opposite in direction. This is really astatement of Newton's third law of motion: 'to everyaction there is an equal and opposite reaction'.

A trolley is propelled by a small carbon dioxide cylinder.As the gas is forced out of the cylinder in one direction,the cylinder, and hence the trolley, are propelled in theother direction. As the trolley accelerates, momentum m vis conserved, and the force m a acting on the trolley at anyinstant is exactly equal in size and opposite in directionto the force acting on the carbon dioxide.

o 2

distance (metres)

ssv

------ --- -- -- ------- ------ -- - - ---

rest on a level road. A camera111 of the speedometer every tworams show the results that, in~ht be obtained.speed varies with time is shown,

I Why has the graph this shaper'"

v=1_5m/s

t=35

speedometer is allowed to runl speed is noted every second. Ai;,peedvaries with time is shownmstant acceleration, this time.nal force acting on the trolley.ion v ~ u + at, when u = O.


The photograph below shows this principle being usedto launch a spacecraft, and in the diagram left, retro-rockets are being fired to slow down a capsule before itenters the Earth's atmosphere.

45

46

A boat is propelled through the water as the propellerpushes the water in the opposite direction.

Something to do1. Stand on roller skates on a smooth level surface and throw some heavyobject away from you. Explain what happens. Why do you not normally observethis result when you throw a ball or even a much heavier object?2. If a bicycle valve is fitted into the stopper of a plastic bottle, a simple 'rocket'can be made. Half fill the bottle with water and pump in air until the cork isforced out. Warning: conduct this experiment out of doors!3. Blow up a toy balloon and throw it into the air with its mouth open. Explainwhat happens.

t

,f 10 gaps between, say, ticksnot the distance between tickse interval between ticks 1 and

dots depends on the speed at;hich tape in the diagrams on ther speed?'!

GRAPHY.ltiflash photograph providesiethods of studying motion. Aar time intervals (left) is used toie, and a time-exposure photo-ely, a camera (below) with a

<isopen shutter can be used to'ing object. In each case a series11.1 regular time intelvals on themages are close together the ob-i} hen they are far apart it is mov-photograph, which was taken inellwhen the tennis racket ismov-ding up and when it is slowing

bout the movement of the ball?'?terval between the images, andotograph can find the distancee the average speed during each

EXPLOSIONSAND,COLLISIONS

There is, of course, no way of recharging the balloon inthe last experiment and it soon comes to rest. In a rocketengine, liquid chemicals are continuously fed underpressure into the combustion chamber. There they burnand produce a steady supply of high-temperature, high-pressure gas. This gas is then ejected from the nozzle ofthe rocket and so propels the rocket in the oppositedirection. The thrust or force exerted on the gas - and soon the rocket - is given by Newton's second law. So farwe have considered the acceleration of a constant mass maccelerated by a constant force F to produce a constantacceleration a.

F = m a = m (.1 v)(.1t)

where .1 v means a 'change of velocity' and .1 t the timeinterval during which the change takes place. In generalthe Greek letter .1 (delta) means 'a small change of'.

Newton's second law is, however, also valid for a chang-ing mass. The force F, in appropriate units, is equal to therate of change of momentum; that is

F = .1 (mv).1t

This change of momentum .1(mv) can result from achange of speed .1 v or a change of mass .1 m, so that

F.1v

= m-.1t

F.1m

or =-v.1t

In the former we consider a constant mass m accelerating.1v d ! h I hanai .1m.A"" an In t e atter a c angmg mass -, moving at a~t .1tconstant speed v. The thrust produced by a rocket motor isequal to the mass of the propellent passing through the

nozzle every second, .1m, multiplied by the velocity v of the.1tgas leaving the nozzle.

47

48

COLLISIONS

Collision damage can be caused by a large mass movingslowly or by a small mass moving very quickly. Noticethe damage caused to a stainless steel plate exposed out-side a spacecraft (Gemini 8) orbiting 400 kilometres abovethe Earth. The damage was caused by a 10-7 g micro-meteorite travelling at about 20 km/ s.

To investigate the damage caused by a car crashing intoa telegraph pole at 50 krn/h, engineers at Cornell Aero-nautical Laboratory dropped a vehicle on to a horizontallymounted pole. From what height must the car be droppedforit to reach that speed?"

MEASURINGMOTION

*

.ethods of timing which do not[time. can be arranged. Thisi~f using the scaler as an elect-rd of a trolley. When the card,ng, interrupts a beam of light,The clock then runs until the

of light.milliseconds, the trolley has

[n 50 milliseconds, which is 20r~onds(0.1s) and therefore 200

erefore 2 metres/ second. Why

-----~ ---

Conservation of momentumWhen two bodies collide, the product mass X velocity isalways the same before and after the collision. We canhowever measure this product only when both bodies arefree to move. The air track (p. 22) enables accuratemeasurements to be taken.


In this photograph a moving vehicle has collided with,and then stuck to, a stationary one. From which side wasthe moving vehicle coming ?36 What can you say about the massof the vehiclesr'? Is this an elastic or inelastic collision?"

An elastic collision between a stationary vehicle and amoving vehicle of the same mass is shown above. Themoving vehicle stops, and the stationary one moves offat the same speed. Momentum is thus conserved.

49


In this photograph two vehicles of different masses andmoving at different speeds collide. They then move apartat different speeds. A shutter mechanism enables us to takea multifiash photograph of the straws (attached to thevehicles) before and after the collision.

The results of this experiment are shown in the abovephotograph. You might like to measure the four speeds andthen work out the total momentum before and after thecollision. Remember that momentum is a vector quantity andthat you must take direction into accounti"

50

L _

MEASURINGMOTION

m is much more difficult thansince it involves speed (dis-

direction. Even if we restrictmeasurement of motion in ameasure the distance travelledl. If it were always possible to'ect we were studying, measur-led. This can be done with cars.rolleys but it becomes rather,with bouncing balls or atomic

.veral techniques which enableintervals fairly accurately, andduced. Here are some of them.

An air gun pellet is fired into a lump of plasticinemounted on an air track vehicle. As the mass of the vehiclewith plasticine can be easily found and its velocity deter-mined from the stop clock reading, the total momentumcan be calculated. If we assume that this momentum isequal to the momentum of the pellet, the speed of a pelletof known mass can be calculated.


The dry ice puck apparatus (p. 24) can be used to studyelastic collisions in two dimensions. The photograph hereshows a multiflash picture in which a moving puck collidedwith a stationary one of equal mass. Use. the photographto compare the momentum before and after the collision.

51

Remember that momentum is a vector quantity. You mustresolve the velocities of the pucks after the collision intocomponents in the original direction and at right ang les to it. 40


This is a cloud chamber photograph of an atomic col-lision between a moving and a stationary particle. What canyou say about the masses of these particles?"

By considering conservation of momentum and mass/energy, nuclear scientists can interpret bubble chamberphotographs such as that illustrated below. Many newparticles have been discovered in this way.

52

1 •••... _

1,

1:

t

s

by a certain amount supportsnical springs stretched by theW, and so on.

ight use a number of identicalo centimetres long. One waymight be to say that one unit ofnne band until it was 15 centi-sing the idea that forces change

::.~~:::<:::::::::]I---_. F= 2 units

• F =3 units

placed side by side, two units of-.d to stretch both bands to 15side by side would exert threeed to 15 centimetres and so on.-lastic bands could therefore bein 'units of force'. In this wayuld be constructed.


at rest

THE PRODUCT Ft

t

speedingup

You have probably stood in a lift and experienced feelingheavier when the lift starts upward or that sinking feelingwhen it starts to go down. Imagine you are standing on aweighing machine in a lift. The pointer will indicate yourweight when the lift is at rest. There are two forces actingon you, the downward pull of the earth and the upwardforce exerted on your feet by the platform of the weighingmachine. As these two forces are equal in magnitude nounbalanced force acts on you, and therefore you do notmove.

When the lift starts to move upward, the platform exertsa greater force on you than it did before. This increased'force is registered by the pointer. As the earth's pullremains the same, there is now an unbalanced force actingon you and thus you accelerate upward.

If the acceleration (a) produced by this unbalanced force(F) changes your speed from u to v in t seconds, we cancalculate the product Ft from Newton's second law.

53

F =ma = m --,-(v_-_u....:...)t

mv - mut

EXPLOSIONSANDCOLLISIONS =? Ft = mv - mu = the change in momentum

In other words, the product Ft is numerically equal to thechange in momentum.

Of course, when the lift stops accelerating and moveswith a steady speed, there will be no unbalanced force onyou. What will the weighing machine show?" What will itshow as the lift decelerates and comes-to rest ?43

Let us consider a typical journey in the lift. Suppose it isat rest for 2 seconds, it accelerates upward for 4 seconds,then moves with a steady speed for 6 seconds, thendecelerates for 3 seconds and finally comes to rest. Sup-pose your weight is 40 kgf or 400 newtons. A graph of thepointer readings on the weighing machine, measured innewtons, might be as shown here.

enCo~! 600enOlCiictI

~ 400Cllc:.coctIEOl 200c:.cOlQi!:

accelerating (up)

steady speed rest

deceleratingI II II II II II II I

I

o 6 8 10 12 14 16 18 20

time (seconds)

2 4

A graph of the unbalanced force acting on you during thesame time would, however, look like this.

enC0

~Cllc

100Cllo.E 0

"0 -100CllucctIm.cc::J

time {seconds)

54

I FORCES I

! 10 N

iged a plastic rod or pen andcer rods or to pick up pieces of"tOWS the dome of a Van deg soap bubbles. The muchator in the photograph is used.celerate atomic particles. Inexerted.

- ~- -

experiments have shown that.h always tends to draw all bitsational attraction. The experi-be used to measure this force.

tcr are, the greater is the attrac-are together, the greater is thes extremely small, a very fine

djacent spheres is measured byseam of light reflected from asuspension wire indicates the

chunks of stone isre apart, it will be

.res apart and

[res apart

would be if they were 4 metrese of force, with the square of therse square law' relationship.


OJo.2

time

-- ----- -----

During the upward acceleration, the area underthe graph(Ft) is 100 X 4 = 400 newton-seconds.

As a = Fim, the acceleration = 100/40 = 2·5 m/s ' and,as the time t = 4 seconds, the final speed v = at = 2 . 5 X 4 =10 m/s-.

Thus the change in momentum = mv = 40 X 10 = 400kg m/s.

The product Ft is numerically equal to the change ofmomentum. The same can be shown when the lift deceler-ates and comes to rest.

IMPULSE

When a force acts for a very short interval of time as, forexample, when you kick a football or strike a golf-ball, arapid change of momentum takes place. But the forceacting is rarely constant. A graph of the force (F) againsttime (t) might look like this.

OJo.2

OJo

2

time

At any moment the product Ft gives the change of momen-tum. For example, at the moment A shown above, theforce is F and in a small interval of time, .1 t, the productF.6.t equals the area under the graph. To find the totalchange in momentum we have to measure the area underthe whole curve.

55

In such circumstances, the total product Ft is often calledthe impulse of the force, or, simply, the impulse.

A given change of momentum will require a certainimpulse. This may be obtained from a large force actingfor a short time or from a small force acting for a longtime. If you try to catch a cricket ball while keeping yourarms rigid, the impulse will last for a short time and theforce exerted on you will be large and painful! If, however,you let your arm move with the ball, the impulse will takelonger but the force exerted will be smaller. The productFt will be the same in each case, since it is equal to thechange of momentum of the ball which finishes at rest ineach case. When you jump from a wall to the ground, youhave momentum just before hitting the ground which ischanged to zero by the impulse. This change would happenquickly if you kept your legs rigid, so that the force wouldbe very great and painful. You usually bend your legs whenyou land so that the time of the impulse is long and theforce is therefore small. The impulse would be the samewhether you kept your legs straight or bent.

When you use a hammer to knock a nail into a plank ofwood, the change of momentum of the hammer takes placequickly, so that a briefbut large force is exerted on the nail.What would happen if the plank were resting on a piece ofsponge rubber as the nail was being hammered?"

Normally when a driver applies his brakes his car comesto rest gradually. A small force acts for a long time. In acollision, however, a much greater force acts for a shorttime. In each case the change of momentum (area underthe force-time graph) is,the same.

At the Road Research laboratory at Crowthorne, carsare crashed into a massive concrete block to investigatethe effects on the driver and passengers. Dummies areplaced in such cars with and without safety belts.


56

I FORCES I

a hovercraft moving on a";m is very much less than thesolid, or solid and liquid, thewill not need to exert a large

! S recently floated on a cushionnetres by a small tractor.

supported and guided by air-able of speeds greater than

e the frictional forces completely,needed to keep the Aerotrain

(ample of a body moving without


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58

These few frames from a high speed filmshowthemotionof a dummy when a car hits a concrete barrier at 60 km/h.This passenger was not using a seat belt! Explain why asafety belt can reduce the injury caused to a passengerinvolved in a car accident:"

---------------

I, either no forces act on it - anor the forces are balanced.

balanced forces keep theconsider the forces acting

photograph as a single force,»rium under the action of threeces are represented by threev;unotice anything special aboutf;·)rces?3

rol on a perfectly level road, itt" has been altered. A force, orFg on the car. What are some of

'ow how motion can be arrest-in a disc brake (left) or by air

liQMMARY This summary includes for completeness a few items notmentioned in the text.1. Forces change either motion i.e. speed or direction, orthe shape of a body2. Forces are exerted by: (a) direct contact includingfriction (b) magnetic fields (c) electric fields (d) gravita-tional fields3. Equations 0.1motion for constant acceleration a

v = u + atv2 = u? + 2 asS -= ut + 1- at?

average velocity _ u + vv=--2

4. Sf units The fundamental SI units used in mechanicsare the metre for length, the kilogramme for mass and thesecond for time.

s

5. Newton's first law A body will stay at rest or continuemoving at a constant speed in a straight line unless anunbalanced force acts on it.6. Newton's second law The acceleration of a body isdirectly proportional to the unbalanced force acting on itand inversely proportional to its mass.

. Fl.e.a =-mkgor

N

59

--.-~'

\ SUMMARY I

60

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7. Newton's third law When two bodies interact theforces they exert on each other are equal in magnitudeand opposite in direction.

Another way of stating the same law is to say that'momentum is conserved in a collision' .

8. Impulse

F=ma=m(v-u)

==> Ft= mv - mu

impulse = change of momentumThe product Ft is called the impulse.9. Gravitational field strength

/N

kg

Field strength g measured in newtons/kilogramme is num-erically the same as the acceleration of gravity g measuredin metres/second".10. Projectiles Vertical and horizontal motion are in-dependent of each other.11. Kinetic energy When a mass m is moving with velocity vit has energy of motion. This is called kinetic energy and itsvalue is tmv2.

I

row it or kick it. In each casethe ball. To make it change

iall.force on the ball. When you\)p its motion by exerting a

\

rces are being used to changethe ball exert a force on you inver to this and to the otherlook will befound on pp. 61-4.)~e, electrons are given off by aby an electric force. The beamforce produced by two large

:0 exert a force on somethingI. lean against a wall, squeeze afd of plasticine, but even thenI[,hrespect to the rest.\}sa tennis ball which has been.is racket. What is the importantif rubber and of plasticine after

IANSWERSI 1. Yes. At any instant, the force you exert on the ball is exactly equal in magnitudeand opposite in direction to the force the ball exerts on you.

2. The rubber quickly returns to its original shape, but the plasticine remains per-manently deformed.

3. The 'lines of action' of these forces pass through a single point. That is, thethree forces are concurrent. This is always the case when a body is in equilibriumunder the action of three forces.

4. There is friction in the car bearings, gears etc. Air resistance will also slow thecar down.

Many people would answer friction between the tyres and the road.' In fact thiswould be a misleading answer although in practice some slipping will occur andsome energy will be transformed to heat as a result of this friction. What wouldhappen if there were no friction between the tyres and the road? What wouldhappen to a car moving on ice?

Friction does in fact stop the car moving, but it is not principally the frictionbetween the tyres and the ground, although it is this friction which causes the wheelsand hence the bearings to rotate. When you are oiling your bicycle wheels where doyou put the oil? On the tyres? On the axle bearings?

5. None. A force would, of course, be needed to get it moving (i.e. cause it toaccelerate) and another force would be needed to slow it down and stop it. How-ever, if there were no friction, no force would be needed to keep it moving oncestarted.

6. Artificial earth satellites keep moving at a steady speed without friction. Stars,planets, moons etc. are other examples.

You might like to puzzle out how it is possible for a satellite to move at aconstant speed round the earth yet be accelerating downwards all the time! Is thereaforce acting on the satellite? Is acceleration a scalar or vector quantity?

F7. 16'

8. If a spring is stretched too far it will not return to its original size when theforce is removed. The greatest force which can be applied without this happeningis called the 'elastic limit'. Hooke's law is not applicable beyond this point.

9. The trolley had moved 10 em during the 50 ms and it could have been acceleratingor decelerating during that time. As average speed is the total distance/total time,we see that it is this quantity that is being measured here. The instantaneous speed,that is the speed at any instant of time, may have varied during the 50 ms period.

10. l.. second.50

11. TapeB.

12. The ball is thrown up into the air and gradually slows down (decelerates). As itfalls it is accelerating. It is then struck by the tennis racket and moves off muchmore quickly at (almost) a constant speed in a straight line.

13. Six images appear above the +-metre stick so that the car took about 6/10second to move a distance of half a metre. It must have been travelling at roughly0.8 m/s

14. The car was accelerating slightly. The distance between the two left-handimages is very slightly greater than the distance between the two right-handimages. If you said it was going at a steady speed you may consider yourself correct.The photograph is not really good enough to detect much acceleration.

61

15. Acceleration.

16. The graph is a straight line because the increase in speed is the same duringeach interval of time. That is, the rate of change of speed (acceleration) is constant.

17. During constant acceleration from rest the speed is directly proportional to thetime (see the diagram on p. 20) but the speed is not proportional to the distanceThe square of the speed is, however, proportional to the distance travelled(V2 = 2as).

18. It is moving at a constant speed.

19. We deduce this from the fact that the images are equally spaced out.

20. This assumes that the flashes themselves have the same time intervals betweenthem.

21. A time exposure photograph could be taken of the white second hand of ablack-faced stop clock. Alternatively the strobe lamp could be used to view aticker-timer vibrating at 50 times per second, or a flywheel rotating at a steadyspeed. The motion would appear to be frozen' if the strobe lamp frequency werethe same as that of the moving body.

ANSWERS

22. As the puck is free to move in two dimensions, this photograph shows thatonce a body is moving it will continue to move at a constant speed in a straightline provided there is no unbalanced force acting on it.

23. The larger engine is capable of exerting a greater force which produces agreater acceleration.

24. The acceleration would be three times the acceleration produced by one elasticthread.

25. The acceleration would be one third of that of the single trolley.

26. The completed table reads

1 N acting on 1kg produces acceleration of 1m/ S23 N " II 1kg II "II 3 m/ S2

F N II 1kg II F m/ S2

F N " 5 kg II E ml S25 .

FN " E mls?m" mkg

I . Facce eratton = Iii

27. A number of known masses can be attached by Sellotape to the end of theblade, and the frequency for each measured. A graph of frequency against mass canthen be plotted.

If an unknown mass were then attached to the end of the blade and the vibrationfrequency measured, the mass could be found from the graph. You may like to tryplotting the period (T) against the mass (m) and also P against m.

28. Yes.

29. Yes.

30. This experiment does not depend on the Earth's gravitational pull. Thisexplains the previous two answers.

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NOTETO THE

\TEACHER

I CONTENTS I

lsentati'es througlwut the world

ationystemms-ording'on of

on

Irateful to the following for per-k photographs: front cover Ford.over Picturepoint Limited; pageleft) Dr. Harold E. Edgerton,hnology; page 5 (right) Stanleytional; page 6 ICI; page 7 (above)and G. Q. Parachute Company

:;ompany Limited; page 8 (above)lin Limited; page 8 (below left):f:S 8 (below right) and 24 (above)rited; page 9 National Physical.ory (Crown copyright reserved);left) Philip Harris Limited; page

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Morris Laboratory InstrumentsAutomation Limited; page 19

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nomena; pages 25 (left), 36 (right)Colour Slides of Physical Pheno-Philip Harris Limited; page 26

.tion Limited; page 45 USIS: pagef~8 and 58 (below) Science Journal;Id Blackett FRS and the RoyalI; pages 57 and 58 Road Researchn copyright reserved).

o Heinemann Educational Booksted for their cooperation.

I ANSWERS I31. Iff represents the force acting on mass m and F represents the force on mass M

f Fa=m=Y

The force must therefore be proportional to the mass ifforce is to be the same inmass

each case. That is, twice the force acts on twice the mass, three times the force onthree times the mass and so on.

32. All these statements with the exception of 7 are correct. The square of thespeed is proportional to the distance (v2 = 2as) and thus the speed is proportionalto the square root of the distance (v = VTciS).

33. The average acceleration of the ball is 1.1 cmt interval! interval0.011 m/to sllo s

= 0.011 X 30 X 30 ml s?

= 9.9 ml s? approx.

34. All these statements except 5 are correct. The vertical speed decreases to zeroat the top of its motion and increases as it falls.

35. s = ~ = (5 X 104)2 X 2 110 = 9.6 metres (approx.)

2a 60 X 60 X

36. From the left. It is moving at half the speed on the right-hand side.

37. As the speed is halved the mass must have doubled if momentum is conserved.Both vehicles must therefore have the same mass.

38. The two vehicles stick together. The collision is therefore inelastic.

39. Ten spaces have been measured in each case.Total momentum before collision = (2 X 6.2) - (3 X 3.6)

12.4 10.81.6 units

Total momentum after collision = (3 X 2.3) - (2 X 2.6)6.9 5.2

1.7 units

40. See diagram on page 64.

41. As the angle formed is 90° the masses of the two particles must be the same.

42. As there is no unbalancedforce, the reading will be the same as it was when youwere at rest.

43. The reading will now be less than it was when you were at rest. Although yourweight is the same the upward force acting on your feet is less as the lift slowsdown.

44. The impulse would last longer as the plank would sink into the rubber. Theforce would therefore be smaller and the nail would not be knocked very far into thewood. The area under the Ft curve would, of course, still be the same, as thechange of momentum of the hammer head would still be the same.

45. When a car is stopped suddenly, for example by running into a brick wall, thepassenger tends to continue moving at the same speed in a straight line - perhapsthrough the windscreen. As the seats are anchored to the floor of the car they willnot move forward. Similarly, if the passenger is wearing seat belts which hold himin the seat he will not be able to continue at the same speed and is thereforeless likely to be seriously injured by being thrown against the windscreen or dash-board.

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LONGMAN PHYSICS TOPICSGeneral Editor J. L. Lewis, Malvern College; formerlyAssociate Organiser, Nuffield O-Ievel Physics Project

This series provides background material for moderncourses in physics. The authors weredosely associatedwith the Nuffield Foundation Physics Project, and thushave an intimate knowledge of its spirit These books arenot textbooks in the conventional sense, nor do they givethe answers to investigations that pupils will be carryingout in the laboratory. Instead they show the relevance andapplication in the outside world of the principles studiedin school.

Documents

Mass in Motion