Cambridge International AS and A Level Mathematics Mechanics

370 Pages·2014·7.58 MB·English

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. Contents
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M1 Mechanics 1 1
Photo credits for CD material: Exercise 14B question 11 © Monkey Business – Fotolia
l = left, c = centre, r = right Chapter 1 Motion in a straight line 2
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Speed and velocity 6
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glad to make suitable arrangements with any copyright holders whom it has not been possible to contact. Acceleration 11
®IGCSE is the registered trademark of University of Cambridge International Examinations. Using areas to find distances and displacements 13
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Chapter 2 The constant acceleration formulae 22
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Setting up a mathematical model 22
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Further examples 31
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. Chapter 3 Forces and Newton’s laws of motion 40
Force diagrams 40
The original MEI author team for Mechanics comprised John Berry, Pat Bryden, Ted Graham, David Holland,
Cliff Pavelin and Roger Porkess. Force and motion 47
Copyright in this format © Roger Porkess and Sophie Goldie, 2012 Pulleys 55
First published in 2012 by Reviewing a mathematical model: air resistance 59
Hodder Education, an Hachette UK company,
338 Euston Road
London NW1 3BH Chapter 4 Applying Newton’s second law along a line 64
Impression number 5 4 3 2 1 Newton’s second law 64
Year 2016 2015 2014 2013 2012
Newton’s second law applied to connected objects 71
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 Chapter 5 Vectors 85
and retrieval system, without permission in writing from the publisher or under licence from
Adding vectors 85
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reproduction) may be obtained from the Copyright Licensing Agency Limited, Saffron Components of a vector 87
House, 6–10 Kirby Street, London EC1N 8TS.
The magnitude and direction of vectors written in component form 92
Cover photo by © Imagestate Media (John Foxx)
Resolving vectors 94
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 2
The language of motion 2
Speed and velocity 6
Acceleration 11
Using areas to find distances and displacements 13
Chapter 2 The constant acceleration formulae 22
Setting up a mathematical model 22
The constant acceleration formulae 24
Further examples 31
Chapter 3 Forces and Newton’s laws of motion 40
Force diagrams 40
Force and motion 47
Pulleys 55
Reviewing a mathematical model: air resistance 59
Chapter 4 Applying Newton’s second law along a line 64
Newton’s second law 64
Newton’s second law applied to connected objects 71
Chapter 5 Vectors 85
Adding vectors 85
Components of a vector 87
The magnitude and direction of vectors written in component form 92
Resolving vectors 94
iii
Chapter 6 Forces in equilibrium and resultant forces 99 Chapter 12 Centre of mass 235
Finding resultant forces 99 Composite bodies 238
Forces in equilibruim 103 Centre of mass for two- and three-dimensional bodies 241
Newton’s second law in two dimensions 114 Sliding and toppling 253
Chapter 7 General motion in a straight line 124 Chapter 13 Uniform motion in a circle 266
Using differentiation 125 Notation 267
Finding displacement from velocity 127 Angular speed 267
The area under a velocity–time graph 128 Velocity and acceleration 271
Finding velocity from acceleration 129 The forces required for circular motion 274
The constant acceleration formulae revisited 131 Examples of circular motion 274
Chapter 8 A model for friction 138 Chapter 14 Hooke’s law 295
A model for friction 139 Strings and springs 296
Modelling with friction 141 Hooke’s law 298
Using Hooke’s law with more than one spring or string 301
Chapter 9 Energy, work and power 154 Work and energy 307
Energy and momentum 154 Vertical motion 314
Work and energy 155
Gravitational potentional energy 163 Chapter 15 Linear motion under a variable force 321
Work and kinetic energy for two-dimensional motion 168 Newton’s second law as a differential equation 322
Power 175 Variable force examples 324
Answers 333
M2 Mechanics 2 183
Index 356
Chapter 10 Motion of a projectile 184
Modelling assumptions for projectile motion 184
Projectile problems 188
Further examples 192
The path of a projectile 202
General equations 203
Chapter 11 Moments of forces 210
Rigid bodies 211
Moments 212
Couples 214
Equilibrium revisited 214
iv
Chapter 12 Centre of mass 235
Composite bodies 238
Centre of mass for two- and three-dimensional bodies 241
Sliding and toppling 253
Chapter 13 Uniform motion in a circle 266
Notation 267
Angular speed 267
Velocity and acceleration 271
The forces required for circular motion 274
Examples of circular motion 274
Chapter 14 Hooke’s law 295
Strings and springs 296
Hooke’s law 298
Using Hooke’s law with more than one spring or string 301
Work and energy 307
Vertical motion 314
Chapter 15 Linear motion under a variable force 321
Newton’s second law as a differential equation 322
Variable force examples 324
Answers 333
Index 356
vv
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
vii
Series Editor
The Cambridge International
AS and A Level Mathematics
syllabus
P2
Cambridge
AS Level
IGCSE P1 S1
Mathematics
Mathematics
M1
M1
S1
S2
A Level
P3
Mathematics
S1
M1
M2
vviiiiii
Mechanics 1
M1