Solutions Manual for Modern Classical Mechanics 1st Edition by T. M. Helliwell, V. V. Sahakian

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Modern Classical Mechanics Solution Manual T. HELLIWELL

V. SAHAKIAN

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Preamble

Our book “Modern Classical Mechanics” includes more than 500 problems, attached as usual in a few pages at the end of every chapter. Some problems are easy – labeled by (*), some are challenging tagged with (***), many others are somewhere in between and are denoted by (**). Truth be told, it took us about as long to collect or invent these problems, solve them, and typeset them, as to write and typeset the main body of the textbook itself.

Some we have assigned already many times in our courses, while others are newly minted.

It is of course through solving problems that most students learn, really learn, the material. And for that to succeed it is obviously important to keep the solutions manual confidential. As every physicist knows, it is only through grappling with problems oneself, making mistakes, following blind alleys, and otherwise challenging oneself, is the material truly learned. Having solutions with which students can compare their work only at the end then becomes an invaluable resource for learning from their mistakes.

In this problem solutions manual we have attempted to solve all of the problems at the end of the chapters. Surprisingly, we have been known to make mistakes. There may even be typos here and there. So we hope that people who use the book and this manual will let us know when they find errors so that we can correct them. We will keep an errata webpage updated at: sahakian.physics.hmc.edu/mcm You can submit corrections and suggestions by contacting sahakian@hmc.edu via email; please include the words “MCM errata” in the subject of the email. We will periodically update the content of the booklet at the Cambridge University Press website.

Finally, we are grateful to Ivy Yuan for help in typesetting this manual.

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Part I

1 Newtonian Particle Mechanics

3

2 Relativity

45

3 The Variational Principle

82

4 Lagrangian Mechanics

108

5 From Classical to Quantum and Back

143

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1

Newtonian Particle Mechanics

1.1 Problems and Solutions

Problem 1.1 A meterstick is at rest in a primed frame of reference, with one end at the origin and the other at x = 1.0 m. (a) Using the Galilean transformation find the location of each end of the stick in the unprimed frame at a particular time t, and then find the length of the meter stick in the unprimed frame. (b) Repeat for the case that the stick is laid out along the positive y axis, with one end at the origin and the other at y = 1.0 m. What is the length of the stick in the unprimed frame?

Solution (a) x = x + vt = x + vt, so the left end has xl = 0 + vt = vt, and the right end has xr = 1.0 m + vt. Therefore the length = xr − xl = 1.0 m. (b) x = 0 for both ends in this case, and y = 0 and y = 1.0 m always. Therefore x = x + vt = vt for both ends, and y = y = 0 and 1.0 m for the two ends, so the length in

the unprimed frame is Δy = Δy = 1.0 m.

Problem 1.2 A river of width D flows uniformly at speed V relative to the shore. A swimmer swims always at speed 2V relative to the water. (a) If the swimmer dives in from one shore and swims in a direction perpendicular to the shoreline in the reference frame of the flowing river, how long does it take her to reach the opposite shore, and how far downstream has she been swept relative to the shore? (b) If instead she wants to swim to a point on the opposite shore directly across from her starting point, at what angle should she swim relative to the direction of the river flow, and how long would it take her to swim across?

Solution (a) Her velocity perpendicular to the shoreline is 2V, so the time to reach the opposite shore D . During this time, she is also swept downstream a distance d = Vt = D2 . is t = 2V (b) She must have an upstream component of velocity V to make up for the river flow.

(2V)2 − V2 = From the Pythagorean theorem, her velocity component across the river is √ 3V and the angle V V 1 θ = tan−1 √ = sin−1 = 30◦ = sin−1 2V 2 3V 3

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