A Level Further Mathematics for OCR A: Mechanics Student Book (AS/A Level)

493 Pages·2018·83.944 MB·English

by  Jess Barker, Nathan Barker, Michele Conway, Janet Such

Brighter Thinking
A Level Further
Mathematics for OCR A
Mechanics Student Book (AS/A Level)
Jess Barker, Nathan Barker, Michele Conway, Janet Such
Course consultant: Vesna Kadelburg
Contents
Introduction
How to use this resource
1 Work, energy and power 1
Section 1: The work done by a force
Section 2: Kinetic energy and the work–energy principle
Section 3: Potential energy, mechanical energy and conservation of mechanical energy
Section 4: Work done by a force at an angle to the direction of motion
Section 5: Power
Mixed practice 1
2 Dimensional analysis
Section 1: Defining and calculating dimensions
Section 2: Units and dimensions of sums, differences and angles
Section 3: Finding dimensions from units and derivatives and predicting formulae
Section 4: Summary of dimensions and units
Mixed practice 2
3 Momentum and collisions 1
Section 1: Momentum and impulse
Section 2: Collisions and the principle of conservation of momentum
Section 3: Restitution, kinetic energy and impulsive tension
Mixed practice 3
4 Circular motion 1
Section 1: Linear speed vs angular speed
Section 2: Acceleration in horizontal circular motion
Section 3: Solving problems involving motion in a horizontal circle
Mixed practice 4
5 Centres of mass 1
Section 1: Centre of mass of a system of point masses
Section 2: Centres of mass of standard shapes
Section 3: Centres of mass of composite bodies
Mixed practice 5
Focus on … Proof 1
Focus on … Problem solving 1
Focus on … Modelling 1
Cross-topic review exercise 1
6 Work, energy and power 2
Section 1: Work done by a variable force
Section 2: Hooke’s law, work done against elasticity and elastic potential energy
Section 3: Problem solving involving work, energy and power
Section 4: Using vectors to calculate work done, kinetic energy and power
Mixed practice 6
7 Linear motion under variable force
Section 1: Working with acceleration, velocity and displacement
Section 2: Variable force
Mixed practice 7
8 Momentum and collisions 2
Section 1: Variable force and vector notation
Section 2: Oblique impacts and the impulse–momentum triangle
Section 3: Oblique collisions of two spheres and impulsive tensions in strings
Mixed practice 8
9 Circular motion 2
Section 1: Conservation of mechanical energy
Section 2: Components of acceleration (a general model)
Section 3: Problem solving situations
Mixed practice 9
10 Centres of mass 2
Section 1: Centres of mass by integration
Section 2: Equilibrium of a rigid body
Mixed practice 10
Focus on … Proof 2
Focus on … Problem solving 2
Focus on … Modelling 2
Cross-topic review exercise 2
AS Level practice paper
A Level practice paper
Formulae
Answers
Worked solution for chapter exercises
1 Work, energy and power 1
2 Dimensional analysis
3 Momentum and collisions 1
4 Circular motion 1
5 Centres of mass 1
6 Work, energy and power 2
7 Linear motion under variable force
8 Momentum and collisions 2
9 Circular motion 2
10 Centres of mass 2
Worked solution for cross topic review exercises
Cross-topic review exercise 1
Cross-topic review exercise 2
AS Level practice paper
A Level practice paper
Glossary
Acknowledgements
Introduction
You have probably been told that mathematics is very useful, yet it can often seem like a lot of techniques
that just have to be learned to answer examination questions. You are now getting to the point where you
will start to see where some of these techniques can be applied in solving real problems. However, as well
as seeing how maths can be useful, we hope that anyone working through this book will realise that it can
also be incredibly frustrating, surprising and ultimately beautiful.
The book is woven around three key themes from the new curriculum:
Proof
Maths is valued because it trains you to think logically and communicate precisely. At a high level, maths is
far less concerned about answers and more about the clear communication of ideas. It is not about being
neat – although that might help! It is about creating a coherent argument that other people can easily
follow but find difficult to refute. Have you ever tried looking at your own work? If you cannot follow it
yourself it is unlikely anybody else will be able to understand it. In maths we communicate using a variety
of means – feel free to use combinations of diagrams, words and algebra to aid your argument. And once
you have attempted a proof, try presenting it to your peers. Look critically (but positively) at some other
people’s attempts. It is only through having your own attempts evaluated and trying to find flaws in other
proofs that you will develop sophisticated mathematical thinking. This is why we have included lots of
common errors in our ’work it out’ boxes – just in case your friends don’t make any mistakes!
Problem solving
Maths is valued because it trains you to look at situations in unusual, creative ways, to persevere and to
evaluate solutions along the way. We have been heavily influenced by a great mathematician and maths
educator, George Polya, who believed that students were not just born with problem solving skills – these
skills were developed by seeing problems being solved and reflecting on the solutions before trying similar
problems. You may not realise it but good mathematicians spend most of their time being stuck. You need
to spend some time on problems you can’t do, trying out different possibilities. If after a while you have not
cracked it then look at the solution and try a similar problem. Don’t be disheartened if you cannot get it
immediately – in fact, the longer you spend puzzling over a problem the more you will learn from the
solution. You may, for example, never need to integrate a rational function in future, but we firmly believe
that the problem solving skills you will develop by trying it can be applied to many other situations.
Modelling
Maths is valued because it helps us solve real-world problems. However, maths describes ideal situations
and the real world is messy! Modelling is about deciding on the important features needed to describe the
essence of a situation and turning that into a mathematical form, then using it to make predictions,
compare to reality and possibly improve the model. In many situations the technical maths is actually the
easy part – especially with modern technology. Deciding which features of reality to include or ignore and
anticipating the consequences of these decisions is the hard part. Yet some fairly drastic assumptions –
such as pretending a car is a single point or that people’s votes are independent – can result in models that
are surprisingly accurate.
More than anything else, this book is about making links. Links between the different chapters, the topics
covered and the themes just discussed, links to other subjects and links to the real world. We hope that
you will grow to see maths as one great complex but beautiful web of interlinking ideas.
Maths is about so much more than examinations, but we hope that if you take on board these ideas (and
do plenty of practice!) you will find maths examinations a much more approachable and possibly even
enjoyable experience. However, always remember that the results of what you write down in a few hours
by yourself in silence under exam conditions is not the only measure you should consider when judging
your mathematical ability – it is only one variable in a much more complicated mathematical model!
How to use this resource
Throughout this resource you will notice particular features that are designed to aid your learning. This
section provides a brief overview of these features.
In this chapter you will learn how to:
calculate the work done by a force
calculate kinetic energy
use the work–energy principle
Learning objectives
A short summary of the content that you will learn in each chapter.
Before you start…
GCSE You should know how to convert 1 Convert metres to
units of distance, speed and kilometres.
time.
A Level Mathematics Student You should know how to 2 Calculate the weight of a car
Book 1 calculate the weight of an object of mass , stating the
from its mass, and know the unit
unit with your answer.
of weight.
Before you start
Points you should know from your previous learning and questions to check that you’re ready to start the
chapter.
WORKED EXAMPLE
The left-hand side shows you how to set out your working. The right-hand side explains the more
difficult steps and helps you understand why a particular method was chosen.
PROOF
Step-by-step walkthroughs of standard proofs and methods of proof.
WORK IT OUT
Can you identify the correct solution and find the mistakes in the two incorrect solutions?
Key point
A summary of the most important methods, facts and formulae.
Explore
Ideas for activities and investigations to extend your understanding of the topic.
Tip
Useful guidance, including on ways of calculating or checking and use of technology.
Each chapter ends with a Checklist of learning and understanding and a Mixed practice exercise,
which includes past paper questions marked with the icon .
In between chapters, you will find extra sections that bring together topics in a more synoptic way.
Focus on…
Unique sections relating to the preceding chapters that develop your skills in proof, problem solving and
modelling.
CROSS-TOPIC REVIEW EXERCISE
Questions covering topics from across the preceding chapters, testing your ability to apply what you have
learned.
You will find AS Level and A Level practice questions towards the end of the resource, as well as a
glossary of key terms (picked out in colour within the chapters), and answers to all questions. Full worked
solutions can be found on the Cambridge Elevate digital platform, along with a digital version of this
Student Resource.
Maths is all about making links, which is why throughout this book you will find signposts emphasising
connections between different topics, applications and suggestions for further research.
Rewind
Reminders of where to find useful information from earlier in your study.
Fast forward
Links to topics that you may cover in greater detail later in your study.
Focus on…
Links to problem solving, modelling or proof exercises that relate to the topic currently being
studied.
Did you know?
Interesting or historical information and links with other subjects to improve your awareness about
how mathematics contributes to society.
Colour-coding of exercises
The questions in the exercises are designed to provide careful progression, ranging from basic fluency to
practice questions. They are uniquely colour-coded, as shown here.
1 A uniform rectangular lamina has vertices at and . Find the coordinates of the
centre of mass of the lamina.
10 Find the increase of elastic potential energy when a light elastic string of natural length and
modulus of elasticity is extended from to .
13 An object of mass is attached to one end of a light elastic string of natural length with its
other end attached to a fixed point, . The modulus of elasticity of the string is . is dropped from
. Find the extension of the string when the object reaches its maximum velocity.
18 A light elastic spring with natural length rests on a smooth horizontal table. One end is attached
to a fixed point and a mass is attached at the other end , held from . The modulus of
elasticity of the spring is .
10 Road surface conditions are being assessed for a horizontal bend in a road that is formed by an arc of a
circle of radious . The road surface could be made of asphalt or concrete. The coefficient of friction
between car tyres and asphalt is , and between car and concrete .
14 A vertical hollow cylinder of radius is rotating about its axis. A particle rough inner surface of the
cylinder. The cylinder and rotate with the same The coefficient of friction between and the cylinder
is .
Black – drill questions. Some of these come in several parts, each with subparts i and ii. You only need
attempt subpart i at first; subpart ii is essentially the same question, which you can use for further practice
if you got part i wrong, for homework, or when you revisit the exercise during revision.
Green – practice questions at a basic level.
Blue – practice questions at an intermediate level.
Red – practice questions at an advanced level.
Purple – challenging questions that apply the concept of the current chapter across other areas of maths.
Yellow – designed to encourage reflection and discussion.
  – indicates content that is for A Level students only
1 Work, energy and power 1
In this chapter you will learn how to:
calculate the work done by a force
calculate kinetic energy
use the work–energy principle
equate gravitational potential energy to work done against gravity
perform calculations using power.
Before you start…
GCSE You should know how to 1 Convert metres to kilometres.
convert units of distance,
speed and time.
A Level You should know how to 2 Calculate the weight of a car of mass
Mathematics calculate the weight of an , stating the unit with your
Student Book 1 object from its mass, and answer.
know the unit of weight.
A Level You should be able to use 3 A resultant force of acts on an
Mathematics Newton’s second law of object of mass . Calculate the
Student Book 1 motion: acceleration of the object.
A Level You should be able to 4 A force of acts on a particle at an
Mathematics resolve a force into angle of to the positive horizontal
Student Book 2 components at right direction. What are the horizontal and
angles to each other. vertical components of the force?
The relationship between work and energy
You have already studied the effect of a force or system of forces in A Level Mathematics.
In this chapter, you will learn the definition of the work done by a force, which is a quantity that is
measured in joules, the same units that are used for energy. You will learn about propulsive and resistive
forces. You will learn about the relationship between work done and two different types of energy: kinetic
energy and gravitational potential energy. You will also learn about power, which is the rate of doing work.
Ideas of work, energy and power are crucial in engineering, enabling engineers to design machines to do
useful work. Hydroelectric power stations work by converting the work done by falling water, first into
kinetic energy as the hydroelectric turbines rotate and then into electricity.
Fast forward
In Chapter 6, you will learn about elastic potential energy and its conversion to kinetic
energy.
Section 1: The work done by a force
Work is done by a force when the object it is applied to moves. The amount of work done is the product of
the force and the distance moved in the direction of the force.
Some forces promote movement, while others resist it. For example, when you cycle into a breeze, your
pedalling promotes movement but the breeze acts against your movement. Forces that promote
movement are called propulsive forces and those that resist movement are known as resistive forces.
Other propulsive forces include the tension in a rope being used to drag an object across the ground and
the driving force of a vehicle engine. The driving force of an engine is often described as its tractive
force. Other resistive forces include friction, vehicle braking and resistance by moving through still air or a
liquid.
Key point 1.1
For a force acting in the direction of motion:
Work done is measured in joules .
, i.e.
For example a force of acting on an object that moves in the direction of the force does
of work. Doubling the force to over the same distance would double the amount of work
done to . Likewise, doubling the distance moved to with an unchanged force of would double
the amount of work done to .
WORKED EXAMPLE 1.1
A box is pushed across a horizontal floor by a horizontal force of . Calculate the work done
by the force.
Use the definition of work done.
State units of work done with your answer.
WORKED EXAMPLE 1.2
A truck driver driving along a horizontal road applies a braking force of for . Calculate the
work done by the brakes, giving your answer in .
Convert to as you need to work in standard units.
Work done by brakes Use the definition of work done.
Change to .
WORKED EXAMPLE 1.3
A crate is lifted by a rope and pulley system. Calculate the work done against gravity.
Apply the definition of work done to the gravitational force. The
force needed to lift the crate is equal to the crate’s weight and