From Fundamentals of Physics
Fundamentals
of Physics
Mechanics, Relativity,
and Thermodynamics
r. s h a n k a r
New Haven and London
Copyright Yale University
From Fundamentals of Physics
Published with assistance from the foundation established in memory of
Amasa Stone Mather of the Class of 1907, Yale College.
c 2014 by Yale University.
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Copyright Yale University
From Fundamentals of Physics
Contents
Preface xiii
1. The Structure of Mechanics
1.1 Introduction and some useful tips
1.2 Kinematics and dynamics
1.3 Average and instantaneous quantities
1.4 Motion at constant acceleration
1.5 Sample problem
1.6 Deriving v2 = v02 + 2a(x − x0 ) using calculus
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2
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2. Motion in Higher Dimensions
2.1 Review
2.2 Vectors in d = 2
2.3 Unit vectors
2.4 Choice of axes and basis vectors
2.5 Derivatives of the position vector r
2.6 Application to circular motion
2.7 Projectile motion
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3. Newton’s Laws I
3.1 Introduction to Newton’s laws of motion
3.2 Newton’s second law
3.3 Two halves of the second law
3.4 Newton’s third law
3.5 Weight and weightlessness
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4. Newton’s Laws II
4.1 A solved example
4.2 Never the whole story
4.3 Motion in d = 2
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From Fundamentals of Physics
viii
Contents
4.4
4.5
4.6
4.7
Friction: static and kinetic
Inclined plane
Coupled masses
Circular motion, loop-the-loop
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5. Law of Conservation of Energy
5.1 Introduction to energy
5.2 The work-energy theorem and power
5.3 Conservation of energy: K2 + U2 = K1 + U1
5.4 Friction and the work-energy theorem
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6. Conservation of Energy in d = 2
6.1 Calculus review
6.2 Work done in d = 2
6.3 Work done in d = 2 and the dot product
6.4 Conservative and non-conservative forces
6.5 Conservative forces
6.6 Application to gravitational potential energy
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7. The Kepler Problem
7.1 Kepler’s laws
7.2 The law of universal gravity
7.3 Details of the orbits
7.4 Law of conservation of energy far from the earth
7.5 Choosing the constant in U
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8. Multi-particle Dynamics
8.1 The two-body problem
8.2 The center of mass
8.3 Law of conservation of momentum
8.4 Rocket science
8.5 Elastic and inelastic collisions
8.6 Scattering in higher dimensions
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9. Rotational Dynamics I
9.1 Introduction to rigid bodies
9.2 Angle of rotation, the radian
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From Fundamentals of Physics
Contents
9.3
9.4
9.5
9.6
ix
Rotation at constant angular acceleration
Rotational inertia, momentum, and energy
Torque and the work-energy theorem
Calculating the moment of inertia
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10. Rotational Dynamics II
10.1 The parallel axis theorem
10.2 Kinetic energy for a general N-body system
10.3 Simultaneous translations and rotations
10.4 Conservation of energy
10.5 Rotational dynamics using τ = dL
dt
10.6 Advanced rotations
10.7 Conservation of angular momentum
10.8 Angular momentum of the figure skater
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11. Rotational Dynamics III
11.1 Static equilibrium
11.2 The seesaw
11.3 A hanging sign
11.4 The leaning ladder
11.5 Rigid-body dynamics in 3d
11.6 The gyroscope
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12. Special Relativity I: The Lorentz Transformation
12.1 Galilean and Newtonian relativity
12.2 Proof of Galilean relativity
12.3 Enter Einstein
12.4 The postulates
12.5 The Lorentz transformation
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13. Special Relativity II: Some Consequences
13.1 Summary of the Lorentz transformation
13.2 The velocity transformation law
13.3 Relativity of simultaneity
13.4 Time dilation
13.4.1 Twin paradox
13.4.2 Length contraction
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From Fundamentals of Physics
x
Contents
13.5 More paradoxes
13.5.1 Too big to fall
13.5.2 Muons in flight
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14. Special Relativity III: Past, Present, and Future
14.1 Past, present, and future in relativity
14.2 Geometry of spacetime
14.3 Rapidity
14.4 Four-vectors
14.5 Proper time
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15. Four-momentum
15.1 Relativistic scattering
15.1.1 Compton effect
15.1.2 Pair production
15.1.3 Photon absorption
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16. Mathematical Methods
16.1 Taylor series of a function
16.2 Examples and issues with the Taylor series
16.3 Taylor series of some popular functions
16.4 Trigonometric and exponential functions
16.5 Properties of complex numbers
16.6 Polar form of complex numbers
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17. Simple Harmonic Motion
17.1 More examples of oscillations
17.2 Superposition of solutions
17.3 Conditions on solutions to the harmonic oscillator
17.4 Exponential functions as generic solutions
17.5 Damped oscillations: a classification
17.5.1 Over-damped oscillations
17.5.2 Under-damped oscillations
17.5.3 Critically damped oscillations
17.6 Driven oscillator
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From Fundamentals of Physics
Contents
xi
18. Waves I
18.1 The wave equation
18.2 Solutions of the wave equation
18.3 Frequency and period
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19. Waves II
19.1 Wave energy and power transmitted
19.2 Doppler effect
19.3 Superposition of waves
19.4 Interference: the double-slit experiment
19.5 Standing waves and musical instruments
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20. Fluids
20.1 Introduction to fluid dynamics and statics
20.1.1 Density and pressure
20.1.2 Pressure as a function of depth
20.2 The hydraulic press
20.3 Archimedes’ principle
20.4 Bernoulli’s equation
20.4.1 Continuity equation
20.5 Applications of Bernoulli’s equation
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21. Heat
21.1 Equilibrium and the zeroth law: temperature
21.2 Calibrating temperature
21.3 Absolute zero and the Kelvin scale
21.4 Heat and specific heat
21.5 Phase change
21.6 Radiation, convection, and conduction
21.7 Heat as molecular kinetic energy
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22. Thermodynamics I
22.1 Recap
22.2 Boltzmann’s constant and Avogadro’s number
22.3 Microscopic definition of absolute temperature
22.4 Statistical properties of matter and radiation
22.5 Thermodynamic processes
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From Fundamentals of Physics
xii
Contents
22.6 Quasi-static processes
22.7 The first law of thermodynamics
22.8 Specific heats: cv and cp
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23. Thermodynamics II
23.1 Cycles and state variables
23.2 Adiabatic processes
23.3 The second law of thermodynamics
23.4 The Carnot engine
23.4.1 Defining T using Carnot engines
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24. Entropy and Irreversibility
24.1 Entropy
24.2 The second law: law of increasing entropy
24.3 Statistical mechanics and entropy
24.4 Entropy of an ideal gas: full microscopic analysis
24.5 Maximum entropy principle illustrated
24.6 The Gibbs formalism
24.7 The third law of thermodynamics
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Index 443
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