Classical Mechanics I (Spring 2019): Homework #3

Due May 16, 2019

[0.5 pt each, total 6 pts, turn in your homework in the class before the class starts]

• By turning in your homework, you acknowledge that you have not received any unpermittedaid, nor have compromised your academic integrity during its preparation. (Remember theSNU College of Natural Sciences Honor Code!)

• Only handwritten answers are accepted except for numerical problems – for which you printout and turn in not just the end results (e.g., plots) but also the source codes.

• For some problems you may want to use an integral table in Appendix E, and/or moreextensive ones found in advanced references such as Zwillinger.

1.-7. Thornton & Marion, Problems 6-11, 6-16, 7-6, 7-10, 7-11, 7-13, 7-18

(Note: For Problem 6-11, first show that the radius of curvature for the parabola is Rc =∣∣∣ dsdφ ∣∣∣ =∣∣∣ (1+y′2)3/2y′′

∣∣∣ where dφ is the angle turned by a line tangential to the curve. For Problem 7-10,

show that your answer in (b) asymptotes to your answer in (a) when m→ 0. For Problem 7-13,you will need to find the line (or angle) about which the small oscillation takes place.)

8. (a) Using Fermat’s Principle to minimize the transit time, prove the law of reflection. Despiteyour urge to immediately reduce the problem into 2-dimension, you will need to explicitly showthat the mid-point along the light path on the mirror, M1, lies in the same plane as P1 and P2.

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(b) Using Fermat’s Principle prove the law of refraction. Again, despite the impression you mayhave been misled to gain by glancing Thornton & Marion Problem 6-7, you will also need toexplicitly show that the mid-point along the light path that is on the planar interface betweenthe media, M2, lies in the same plane as P1 and P2.

9. (a) Dido problem of isoperimetry. To tackle this problem, first study the case of the calculusof variations where the equation of constraint is written in an integral form (Section 6.6 of yourtextbook). Prove Eq.(6.78) if you desire. Then work out Example 6.6. If you think there is atypo in the textbook, highlight it clearly in your answer.

(b) Now solve the problem differently by finding the functional in the form of∫f{y, y′} ds where

s is the distance along the curve y.

10. Show that the shortest path between two given points on the surface of a sphere is a greatcircle – in four different ways below.

(a) By introducing a functional in the form of∫f1{θ, θ′} dφ in spherical coordinates.

(b) By using an alternative functional form of∫f2{θ, φ′} dθ.

(c) By rewriting the functional in the form of∫f3{y′, z′} dx in Cartesian coordinates, and

employing the method of Euler equations with an auxiliary condition imposed.

(d) By rewriting the functional in the form of∫f4{x′, y′, z′} ds where s denotes a variable chosen

for convenient parametrization for x, y, z – such as the distance along the path.

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11. A yo-yo can be modeled as two uniform disks of mass m/2 and radius R1 with a weightlessaxle of radius R2 in between. The thickness of the axle can be neglected. A massless string issuspended vertically from a fixed point, and the other end is wrapped multiple times around theaxle. When the yo-yo is released from the rest vertically down, it rotates as the string unwinds.

(a) Find the yo-yo’s acceleration using the Newtonian method.

(b) Repeat (a) using the Lagrangian method.

12. Consider a simple pendulum whose point of support moves on a rotating massless rim, asdepicted in Example 7.5.

(a) Work out Example 7.5.

(b) Solve the resulting nonlinear equation of motion numerically. Check if θ(t) resembles thesinusoidal oscillations of a linear simple pendulum if the rim is sufficiently small and rotatesslowly. Assume θ(0) = 0.2 and θ̇(0) = 0. For convenience, plot the solution for 0 < t < 100with a = 0.2, b = 1, g = 1, and ω = 0.1. Experiment other values and ranges if you want.

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