A CALCULUS BASED INTRODUCTION TO MECHANICS

183 Pages·020.385 MB·English

by  CPAC

PHYS 110 Textbook
A Calculus Based Introduction to Mechanics
c R. Keeler and M. Laidlaw
(cid:13)
UVic 2017
ii
Contents
Preface ix
1 Vectors 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Displacement - a motivating example . . . . . . . . . . . . . . 1
1.3 Vectors and their properties . . . . . . . . . . . . . . . . . . . 5
1.3.1 Coordinate systems . . . . . . . . . . . . . . . . . . . 6
1.4 Manipulation of Vectors . . . . . . . . . . . . . . . . . . . . . 6
1.5 Scalar Product . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Vector Product . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.7 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . 12
1.7.2 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Translational Equilibrium 15
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Particle Model . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Newton’s First Law. . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Particles in Equilibrium . . . . . . . . . . . . . . . . . . . . . 17
2.4.1 Equilibrium Example . . . . . . . . . . . . . . . . . . 18
2.5 Newton’s Third Law . . . . . . . . . . . . . . . . . . . . . . . 21
2.5.1 Two masses interacting . . . . . . . . . . . . . . . . . 22
2.6 Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6.1 Gravity near the Earth’s surface . . . . . . . . . . . . 24
2.6.2 Force provided by a rope . . . . . . . . . . . . . . . . 24
2.6.3 Normal Force . . . . . . . . . . . . . . . . . . . . . . . 25
2.6.4 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.6.5 Friction example . . . . . . . . . . . . . . . . . . . . . 26
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3 Rotational Equilibrium 29
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Rigid Body Model . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Conditions for Rotational Equilibrium . . . . . . . . . . . . . 30
3.3.1 Independence of torque on origin in the case of net
zero force . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3.2 Example of rotational equilibrium . . . . . . . . . . . 32
3.4 Centres of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4.1 Revisiting rotational equilibrium with centre of mass . 37
3.5 Two examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.5.1 A vertical rod . . . . . . . . . . . . . . . . . . . . . . . 38
3.5.2 Torque considering three dimensions . . . . . . . . . . 40
4 Differential Calculus 43
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Graphical Representations of functions . . . . . . . . . . . . . 43
4.3 Derivatives and Slopes . . . . . . . . . . . . . . . . . . . . . . 44
4.4 Calculating Derivatives. . . . . . . . . . . . . . . . . . . . . . 46
4.4.1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 47
4.4.2 Product rule and Chain rule. . . . . . . . . . . . . . . 48
4.5 The exponential function and the logarithm . . . . . . . . . . 50
4.6 Interlude: complex numbers . . . . . . . . . . . . . . . . . . . 51
4.7 Sine and Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.8 Advanced topic: Taylor polynomials . . . . . . . . . . . . . . 54
5 Kinematics 57
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2 Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3 Space and Time . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3.1 Average Velocity . . . . . . . . . . . . . . . . . . . . . 60
5.4 Instantaneous Velocity . . . . . . . . . . . . . . . . . . . . . . 60
5.5 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.6 Motion in One Dimension . . . . . . . . . . . . . . . . . . . . 64
5.6.1 Projectile Motion with Gravity . . . . . . . . . . . . . 66
5.7 Projectile Motion . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.8 Motion in a Circle . . . . . . . . . . . . . . . . . . . . . . . . 68
6 Newton’s Second Law 73
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.2 Statement of the Second Law . . . . . . . . . . . . . . . . . . 73
CONTENTS v
6.3 Examples of application of Newton’s Second Law . . . . . . . 74
6.3.1 A block sliding up a rough slope . . . . . . . . . . . . 74
6.3.2 Masses in contact . . . . . . . . . . . . . . . . . . . . . 76
6.3.3 Atwood machine . . . . . . . . . . . . . . . . . . . . . 78
6.3.4 Conical pendulum . . . . . . . . . . . . . . . . . . . . 80
7 Forces 83
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.2 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.2.1 Star orbiting another star . . . . . . . . . . . . . . . . 85
7.2.2 Lagrange Points . . . . . . . . . . . . . . . . . . . . . 87
7.3 Electric Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.3.1 Electric Field . . . . . . . . . . . . . . . . . . . . . . . 90
7.4 Lorentz Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.4.1 Circular motion in a magnetic field . . . . . . . . . . . 91
7.4.2 Velocity Selector . . . . . . . . . . . . . . . . . . . . . 93
7.5 Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.6 Pressure and Buoyancy . . . . . . . . . . . . . . . . . . . . . 95
7.6.1 Bouyancy . . . . . . . . . . . . . . . . . . . . . . . . . 97
8 Integral Calculus 101
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
8.2 Enclosed area and Riemann sums . . . . . . . . . . . . . . . . 101
8.3 Fundamental Theorem of Calculus . . . . . . . . . . . . . . . 104
8.4 ExamplesapplyingtheFundamentalTheoremtosimplefunc-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.4.1 Constant functions . . . . . . . . . . . . . . . . . . . . 107
8.4.2 Linear functions . . . . . . . . . . . . . . . . . . . . . 108
8.4.3 Quadratic Function . . . . . . . . . . . . . . . . . . . 110
8.4.4 Portion of a circle . . . . . . . . . . . . . . . . . . . . 112
8.5 Application of Integral Calculus to Physics problems . . . . . 113
8.5.1 Mass of a cylindrical solid . . . . . . . . . . . . . . . . 114
8.5.2 Displacement given non-constant velocity . . . . . . . 115
9 Momentum 117
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
9.2 Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
9.3 Conservation of Momentum . . . . . . . . . . . . . . . . . . . 118
9.4 Impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.5 Centre of Mass Frame . . . . . . . . . . . . . . . . . . . . . . 122
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9.5.1 Zero Momentum Frame . . . . . . . . . . . . . . . . . 123
9.5.2 Centre of Mass Position . . . . . . . . . . . . . . . . . 124
9.5.3 Nonzero Acceleration . . . . . . . . . . . . . . . . . . 126
9.6 Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
9.6.1 Galilean Transformation . . . . . . . . . . . . . . . . . 127
10 Angular Momentum 129
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
10.2 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
10.3 Conservation of Angular Momentum . . . . . . . . . . . . . . 130
10.4 Rigid Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
10.5 One dimensional rotational kinematics . . . . . . . . . . . . . 134
10.6 Examples - Atwood machine with massive pulley . . . . . . . 135
10.7 Parallel Axis Theorem . . . . . . . . . . . . . . . . . . . . . . 137
10.8 Rolling Objects with constant speed . . . . . . . . . . . . . . 139
11 Work and Energy 141
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
11.2 Mechanical Work with constant Forces . . . . . . . . . . . . . 141
11.3 Forces that vary with position . . . . . . . . . . . . . . . . . . 144
11.4 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 146
11.5 Energy and work for a rotating rigid body . . . . . . . . . . . 147
11.6 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
11.7 Collisions and Scattering . . . . . . . . . . . . . . . . . . . . . 148
11.7.1 Elastic Scattering . . . . . . . . . . . . . . . . . . . . . 148
11.7.2 Totally Inelastic Scattering . . . . . . . . . . . . . . . 152
11.7.3 Inelastic Scattering . . . . . . . . . . . . . . . . . . . . 153
12 Potential Energy 155
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
12.2 Work-Energy Theorem Revisited . . . . . . . . . . . . . . . . 155
12.2.1 Work done by a varying force - procedure . . . . . . . 155
12.2.2 Work done by varying force - example . . . . . . . . . 156
12.2.3 Conservative and Non-Conservative forces . . . . . . . 160
12.2.4 Work-Energy theorem and potential energy . . . . . . 161
12.3 Constant forces . . . . . . . . . . . . . . . . . . . . . . . . . . 163
12.3.1 Gravity near the Earth’s surface . . . . . . . . . . . . 163
12.3.2 Constant Electric Field . . . . . . . . . . . . . . . . . 164
12.3.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 164
12.4 Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
CONTENTS vii
12.4.1 Example - Spring and friction . . . . . . . . . . . . . . 167
12.5 Gravity and Electrostatic forces . . . . . . . . . . . . . . . . . 169
12.5.1 The integral of 1 rˆ . . . . . . . . . . . . . . . . . . . . 169
r2
12.5.2 Potential Energy for the Coulomb force . . . . . . . . 171
12.5.3 Potential Energy for Newtonian Gravity . . . . . . . . 171
12.5.4 Example - Escape Speed and Orbital Energy . . . . . 172
viii CONTENTS
Preface
A student entering the first year of a university Science or Engineering pro-
gram is typically expected to undertake a multiple-term calculus-based sur-
vey of physics. This survey is usually accompanied by two terms worth of
single-variable differential and integral calculus, and often either accompa-
nied or followed by courses on multi-variable calculus, linear algebra, and
differential equations. The topics covered in the survey of physics can vary
frominstitutiontoinstitution,buttheyalmostinvariablystartwithasingle-
term introduction to mechanics. This is a text intended for such a course.
Thecourseforwhichthistextisdesignedhasasitsintendedprerequisites
a high school level algebra-based survey of physics, and a similar level pre-
calculuscourse. Itisintendedtobetakenconcurrentlywithanintroductory
calculus course. As there are several mathematical topics that are essential
totheunderstandingofmechanics,thistextincludesbriefandself-contained
introductions to vector algebra, differentiation, and integration.
The choices that we have made for topics and order of presentation
deviatefrommanyoftheintroductorytextsthatareavailable. Inparticular,
we open with a discussion of vectors and then apply this concept directly
to translational and rotational equilibrium. This is done in part because
our experience is that manipulating vectors is novel for our students, and in
need of reinforcement. It is also done in part because we expect the course
to be taken concurrently with a calculus course and the typical introductory
calculuscoursetakesalittlewhiletointroducederivatives. Ourdevelopment
of acceleration and velocity only touches briefly on one-dimensional motion,
because of the time we spent developing vectors. We include discussions of
Newtoniangravity,theCoulombforce,andtheLorentzforce. Ourtreatment
of work and energy is vector-calculus based.
We intend the text to be rigorous enough that a student who completes
a course based on this will be well prepared for intermediate-level courses
in physics, and for courses which build on the knowledge from first-year
physics. A guiding principle in our development of the text is that we do
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not say anything which will need to be corrected in future courses.
Thistextevolvedfromlecturenotesassociatedwithintroductoryphysics
coursestaughtbytheauthorsattheUniversityofVictoriaintheearly2010s.
The authors would like to thank our dean of science and our department
chair for their support and encouragement in this project. They would also
like to express their appreciation for their students who asked the questions
and made the comments that sparked them to write the text.
Mark Laidlaw
Richard Keeler
Victoria, BC