Cambridge International AS and A Level Mathematics Mechanics PDF

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Cambridge International AS and A Level Mathematics Mechanics Sophie Goldie Series Editor: Roger Porkess Questions from the Cambridge International Examinations AS and A Level Mathematics papers are reproduced by permission of University of Cambridge International Examinations. Questions from the ME...

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Cambridge

International AS and A Level Mathematics

Mechanics Sophie Goldie Series Editor: Roger Porkess

Questions from the Cambridge International Examinations AS and A Level Mathematics papers are reproduced by permission of University of Cambridge International Examinations. Questions from the MEI AS and A Level Mathematics papers are reproduced by permission of OCR. We are grateful to the following companies, institutions and individuals who have given permission to reproduce photographs in this book. Photo credits: page 2 © Mathematics in Education and Industry; page 6 © Radu Razvan – Fotolia; page 22 © Iain Masterton / Alamy; page 24 © photoclicks – Fotolia; page 40 © DOD Photo / Alamy; page 44 © Dr Jeremy Burgess / Science Photo Library; page 45 ©Jonathan Pope / http://commons.wikimedia.org/wiki/File:Olympic_Curling,_Vancouver_2010_crop_sweeping.jpg/http:// creativecommons.org/licenses/by/2.0/deed.en/16thJan2011; page 47 © imagehit – Fotolia; page 53 © NASA / Goddard Space Flight Center / Arizona State University; page 59 © Lebrecht Music and Arts Photo Library / Alamy; page 60 © Mehau Kulyk / Science Photo Library; page 64 © Dmitry Lobanov – Fotolia; page 67 © Imagestate Media (John Foxx); page 85 l © Dean Moriarty – Fotolia; page 85 c © Masson – Fotolia; page 85 r © Marzanna Syncerz – Fotolia; page 99 ©Tifonimages – Fotolia; page 115 © Kathrin39 – Fotolia; page 138 © SHOUT / Alamy; page 153 © Mathematics in Education and Industry; page 154 © Image Asset Management Ltd. / SuperStock; page 175 © Steve Mann – Fotolia; page 184 © photobyjimshane – Fotolia; page 253 © Millbrook Proving Ground Ltd; page 264 © cube197 – Fotolia; page 266 l © M.Rosenwirth – Fotolia; page 266 c ©Michael Steele / Getty Images; page 266 r © NickR – Fotolia; page 280 © Steeve ROCHE – Fotolia; page 295 © Lovrencg – Fotolia; page 308 © blueee – Fotolia Photo credits for CD material: Exercise 14B question 11 © Monkey Business – Fotolia l = left, c = centre, r = right All designated trademarks and brands are protected by their respective trademarks. Every effort has been made to trace and acknowledge ownership of copyright. The publishers will be glad to make suitable arrangements with any copyright holders whom it has not been possible to contact. ®IGCSE is the registered trademark of University of Cambridge International Examinations. Hachette UK’s policy is to use papers that are natural, renewable and recyclable products and made from wood grown in sustainable forests. The logging and manufacturing processes are expected to conform to the environmental regulations of the country of origin. Orders: please contact Bookpoint Ltd, 130 Milton Park, Abingdon, Oxon OX14 4SB. Telephone: (44) 01235 827720. Fax: (44) 01235 400454. Lines are open 9.00–5.00, Monday to Saturday, with a 24-hour message answering service. Visit our website at www.hoddereducation.co.uk Much of the material in this book was published originally as part of the MEI Structured Mathematics series. It has been carefully adapted for the Cambridge International AS and A Level Mathematics syllabus. The original MEI author team for Mechanics comprised John Berry, Pat Bryden, Ted Graham, David Holland, Cliff Pavelin and Roger Porkess. Copyright in this format © Roger Porkess and Sophie Goldie, 2012 First published in 2012 by Hodder Education, an Hachette UK company, 338 Euston Road London NW1 3BH Impression number 5 4 3 2 1 Year 2016 2015 2014 2013 2012 All rights reserved. Apart from any use permitted under UK copyright law, no part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or held within any information storage and retrieval system, without permission in writing from the publisher or under licence from the Copyright Licensing Agency Limited. Further details of such licences (for reprographic reproduction) may be obtained from the Copyright Licensing Agency Limited, Saffron House, 6–10 Kirby Street, London EC1N 8TS. Cover photo by © Imagestate Media (John Foxx) Illustrations by Pantek Media, Maidstone, Kent Typeset in 10.5pt Minion by Pantek Media, Maidstone, Kent Printed in Dubai A catalogue record for this title is available from the British Library ISBN 978 1444 14648 6

This eBook does not include the ancillary media that was packaged with the printed version of the book.

Contents Key to symbols in this book vi Introduction vii The Cambridge International AS and A Level Mathematics syllabus viii

M1 Mechanics 1

1

Chapter 1

Motion in a straight line The language of motion Speed and velocity Acceleration Using areas to find distances and displacements

2 2 6 11 13

Chapter 2

The constant acceleration formulae Setting up a mathematical model The constant acceleration formulae Further examples

22 22 24 31

Chapter 3

Forces and Newton’s laws of motion Force diagrams Force and motion Pulleys Reviewing a mathematical model: air resistance

40 40 47 55 59

Chapter 4

Applying Newton’s second law along a line Newton’s second law Newton’s second law applied to connected objects

64 64 71

Chapter 5

Vectors Adding vectors Components of a vector The magnitude and direction of vectors written in component form Resolving vectors

85 85 87 92 94

iii

Chapter 6

Forces in equilibrium and resultant forces Finding resultant forces Forces in equilibruim Newton’s second law in two dimensions

99 99 103 114

Chapter 7

General motion in a straight line Using differentiation Finding displacement from velocity The area under a velocity–time graph Finding velocity from acceleration The constant acceleration formulae revisited

124 125 127 128 129 131

Chapter 8

A model for friction A model for friction Modelling with friction

138 139 141

Chapter 9

Energy, work and power Energy and momentum Work and energy Gravitational potentional energy Work and kinetic energy for two-dimensional motion Power

154 154 155 163 168 175

M2 Mechanics 2

iv

183

Chapter 10

Motion of a projectile Modelling assumptions for projectile motion Projectile problems Further examples The path of a projectile General equations

184 184 188 192 202 203

Chapter 11

Moments of forces Rigid bodies Moments Couples Equilibrium revisited

210 211 212 214 214

Chapter 12

Centre of mass Composite bodies Centre of mass for two- and three-dimensional bodies Sliding and toppling

235 238 241 253

Chapter 13

Uniform motion in a circle Notation Angular speed Velocity and acceleration The forces required for circular motion Examples of circular motion

266 267 267 271 274 274

Chapter 14

Hooke’s law Strings and springs Hooke’s law Using Hooke’s law with more than one spring or string Work and energy Vertical motion

295 296 298 301 307 314

Chapter 15

Linear motion under a variable force Newton’s second law as a differential equation Variable force examples

321 322 324

Answers Index

333 356

v

Key to symbols in this book ? ●

This symbol means that you may want to discuss a point with your teacher. If you are working on your own there are answers in the back of the book. It is important, however, that you have a go at answering the questions before looking up the answers if you are to understand the mathematics fully.

! This is a warning sign. It is used where a common mistake, misunderstanding or tricky point is being described. This is the ICT icon. It indicates where you could use a graphic calculator or a computer. Graphical calculators and computers are not permitted in any of the examinations for the Cambridge International AS and A Level Mathematics 9709 syllabus, however, so these activities are optional. This symbol and a dotted line down the right-hand side of the page indicates material which is beyond the syllabus but which is included for completeness.

vi

Introduction This is one of a series of books for the University of Cambridge International Examinations syllabus for Cambridge International AS and A Level Mathematics 9709. There are fifteen chapters in this book; the first nine cover Mechanics 1 and the remaining six Mechanics 2. The series also includes two books for pure mathematics and one for statistics. These books are based on the highly successful series for the Mathematics in Education and Industry (MEI) syllabus in the UK but they have been redesigned for Cambridge International students; where appropriate new material has been written and the exercises contain many past Cambridge examination questions. An overview of the units making up the Cambridge international syllabus is given in the diagram on the next page. Throughout the series the emphasis is on understanding the mathematics as well as routine calculations. The various exercises provide plenty of scope for practising basic techniques; they also contain many typical examination questions. In the examinations of the Cambridge International AS and A Level Mathematics 9709 syllabus the value of g is taken to be 10 m s−2 and this convention is used in this book; however, in a few questions readers are introduced to a more accurate value, typically 9.8 m s−2. An important feature of this series is the electronic support. There is an accompanying disc containing two types of Personal Tutor presentation: examination-style questions, in which the solutions are written out, step by step, with an accompanying verbal explanation, and test yourself questions; these are multiple-choice with explanations of the mistakes that lead to the wrong answers as well as full solutions for the correct ones. In addition, extensive online support is available via the MEI website, www.mei.org.uk. The books are written on the assumption that students have covered and understood the work in the Cambridge IGCSE® syllabus. There are places where the books show how the ideas can be taken further or where fundamental underpinning work is explored and such work is marked as ‘Extension’. The original MEI author team would like to thank Sophie Goldie who has carried out the extensive task of presenting their work in a suitable form for Cambridge international students and for her original contributions. They would also like to thank University of Cambridge International Examinations for their detailed advice in preparing the books and for permission to use many past examination questions. Roger Porkess Series Editor

vii

The Cambridge International AS and A Level Mathematics syllabus P2 Cambridge IGCSE Mathematics

P1

S1

AS Level Mathematics

M1

S1

M1 S2

P3 M1

viii

S1 M2

A Level Mathematics

Mechanics 1

M1

Motion in a straight line

M1  1

1

Motion in a straight line The whole burden of philosophy seems to consist in this – from the phenomena of motions to investigate the forces of nature. Isaac Newton

The language of motion Throw a small object such as a marble straight up in the air and think about the words you could use to describe its motion from the instant just after it leaves your hand to the instant just before it hits the floor. Some of your words might involve the idea of direction. Other words might be to do with the position of the marble, its speed or whether it is slowing down or speeding up. Underlying many of these is time. Direction

The marble moves as it does because of the gravitational pull of the earth. We understand directional words such as up and down because we experience this pull towards the centre of the earth all the time. The vertical direction is along the line towards or away from the centre of the earth. In mathematics a quantity which has only size, or magnitude, is called a scalar. One which has both magnitude and a direction in space is called a vector. Distance, position and displacement

The total distance travelled by the marble at any time does not depend on its direction. It is a scalar quantity. 2

Position and displacement are two vectors related to distance: they have direction as well as magnitude. Here their direction is up or down and you decide which of these is positive. When up is taken to be positive, down is negative.

top position +1.25m

positive direction

1.25 m

The language of motion

The position of the marble is then its distance above a fixed origin, for example the distance above the place it first left your hand.

M1 1

hand zero position

Figure 1.1 

When it reaches the top, the marble might have travelled a distance of 1.25 m. Relative to your hand its position is then 1.25 m upwards or +1.25 m. At the instant it returns to the same level as your hand it will have travelled a total distance of 2.5 m. Its position, however, is zero upwards. A position is always referred to a fixed origin but a displacement can be measured from any position. When the marble returns to the level of your hand, its displacement is zero relative to your hand but −1.25 m relative to the top.

? ●

What are the positions of the particles A, B and C in the diagram below? A –4

B –3

–2

–1

0

C 1

2

3

4

5

x

Figure 1.2 



What is the displacement of B



(i) 

relative to A       (ii)  relative to C?

Diagrams and graphs

In mathematics, it is important to use words precisely, even though they might be used more loosely in everyday life. In addition, a picture in the form of a diagram or graph can often be used to show the information more clearly. 3

M1  1 Motion in a straight line

Figure 1.3 is a diagram showing the direction of motion of the marble and relevant distances. The direction of motion is indicated by an arrow. Figure 1.4 is a graph showing the position above the level of your hand against the time. Notice that it is not the path of the marble.

position (metres)

A

A

1.25 m

H

2

1

B

0

0.5

1

B

1.5

time (s)

1m –1

C

Figure 1.3 

? ●

C

Figure 1.4 

The graph in figure 1.4 shows that the position is negative after one second (point B). What does this negative position mean?

Note When drawing a graph it is very important to specify your axes carefully. Graphs showing motion usually have time along the horizontal axis. Then you have to decide where the origin is and which direction is positive on the vertical axis. In this graph the origin is at hand level and upwards is positive. The time is measured from the instant the marble leaves your hand.

Notation and units

As with most mathematics, you will see in this book that certain letters are commonly used to denote certain quantities. This makes things easier to follow. Here the letters used are:

4

●●

s, h, x, y and z for position

●●

t for time measured from a starting instant

●●

u and v for velocity

●●

a for acceleration.

The S.I. (Système International d’Unités) unit for distance is the metre (m), that for time is the second (s) and that for mass the kilogram (kg). Other units follow from these so speed is measured in metres per second, written m s−1. 1

 hen the origin for the motion of the marble (see figure 1.3) is on the W ground, what is its position (i) (ii)

2

when it leaves your hand? at the top?

Exercise 1A

EXERCISE 1A

M1 1

A boy throws a ball vertically upwards so that its position y m at time t is as shown in the graph. position (m) 7 6 5 4 3 2 1 0

1

2 t time (s)

Write down the position of the ball at times t = 0, 0.4, 0.8, 1.2, 1.6 and 2. Calculate the displacement of the ball relative to its starting position at these times. (iii) What is the total distance travelled (a)  during the first 0.8 s      (b)  during the 2 s of the motion? (i)

(ii)

3

The position of a particle moving along a straight horizontal groove is given by x = 2 + t(t − 3) for 0  t  5 where x is measured in metres and t in seconds. What is the position of the particle at times t = 0, 1, 1.5, 2, 3, 4 and 5? (ii) Draw a diagram to show the path of the particle, marking its position at these times. (iii) Find the displacement of the particle relative to its initial position at t = 5. (iv) Calculate the total distance travelled during the motion. (i)

4

For each of the following situations sketch a graph of position against time. Show clearly the origin and the positive direction. A stone is dropped from a bridge which is 40 m above a river. (ii) A parachutist jumps from a helicopter which is hovering at 2000 m. She opens her parachute after 10 s of free fall. (iii) A bungee jumper on the end of an elastic string jumps from a high bridge. (i)

5

Motion in a straight line

M1  1



5

 he diagram is a sketch of T the position–time graph for a fairground ride. (i)

(ii)

position

Describe the motion, stating in particular what happens at O, A, B, C and D. What type of ride is this?

A B

O

D time C

Speed and velocity Speed is a scalar quantity and does not involve direction. Velocity is the vector related to speed; its magnitude is the speed but it also has a direction. When an object is moving in the negative direction, its velocity is negative.

Amy has to post a letter on her way to college. The post box is 500 m east of her house and the college is 2.5 km to the west. Amy cycles at a steady speed of 10 m s−1 and takes 10 s at the post box to find the letter and post it. Figure 1.5 shows Amy’s journey using east as the positive direction. The distance of 2.5 km has been changed to metres so that the units are consistent. C

10 m s 2500 m

1

H

P 500 m

east positive direction

Figure 1.5 

6

After she leaves the post box Amy is travelling west so her velocity is negative. It is −10 m s−1.

The distances and times for the three parts of Amy’s journey are: position (m) 1000

Time

500 m

500 = 50 s 10

500

At post box

0 m

10 s

– 500

Post box to college

3000 m

3000 = 300 s 10

Home to post box

0 – 1000

50 100 150 200 250 300 350 time (s)

– 1500 – 2000

These can be used to draw the position–time graph using home as the origin, as in figure 1.6.

? ●

– 2500

M1 1 Speed and velocity

Distance

Figure 1.6 

Calculate the gradient of the three port...