HW5 - 1 = 2 = 3 = . Write down the Lagrangian of the system...

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Physics 110 Intermediate Mechanics HW#5, due May 19 th 2010 1. A point mass is under the influence of gravitational attraction from M, which is much more massive. At t =0 the position and velocity of m is as shown. a. When E=0 , write down the full orbit as a function of ? , ± , ² 0 . Convince yourself that the escape velocity is independent of ² 0 (15 pts). b. Show that if a set of initial parameters ? , ± , ² 0 produce a bound orbit, another set of parameters ? , ± , ³ − ² 0 also result in a bound orbit with the same orbital period . What is that orbital period? (15 pts) . 2. Coupled oscillations (30 pts): use ´ 1 , ´ 2 , ´ 3 as the generalized coordinates to describe a triple pendulum, as shown. a. µ 1 = µ 2 = µ 3 = µ ;
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Unformatted text preview: 1 = 2 = 3 = . Write down the Lagrangian of the system (6 pts). b. Approximate the Lagrangian in the small angle limit 1 , 2 , 3 <<1, to the leading order involving 1 , 2 , 3 (6 pts) c. Write down the Euler-Lagrange equations. (6 pts) d. Find the normal frequencies of the system. (6 pts) e. Find the normal modes of the system. You dont need to normalize the vectors, but please illustrate schematically what they look like (6 pts). f. Initially the system is at rest and 1 1 << = . Use the normal modes to solve the time evolution of 1 , 2 , 3 (6 pts). 2 1 1 l 2 l g 1 m 2 m 3 3 l 3 m...
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This note was uploaded on 02/01/2011 for the course MATH 171 taught by Professor R during the Spring '09 term at Stanford.

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