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

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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 𝑅𝑅 , 𝑣𝑣 , 𝜃𝜃 0 . Convince yourself that the escape velocity is independent of 𝜃𝜃 0 (15 pts). b. Show that if a set of initial parameters 𝑅𝑅 , 𝑣𝑣 , 𝜃𝜃 0 produce a bound orbit, another set of parameters 𝑅𝑅 , 𝑣𝑣 , 𝜋𝜋 − 𝜃𝜃 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 𝜙𝜙 1 , 𝜙𝜙 2 , 𝜙𝜙 3 as the generalized coordinates to describe a triple pendulum, as shown. a. 𝑙𝑙 1 = 𝑙𝑙 2 = 𝑙𝑙
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Unformatted text preview: Â¶ 1 = Â¶ 2 = Â¶ 3 = Â¶ . Write down the Lagrangian of the system (6 pts). b. Approximate the Lagrangian in the small angle limit Â´ 1 , Â´ 2 , Â´ 3 <<1, to the leading order involving Â´ 1 , Â´ 2 , Â´ 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 donâ€™t 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 << = Î³ Ï† . Use the normal modes to solve the time evolution of Â´ 1 , Â´ 2 , Â´ 3 (6 pts). 2 1 1 l 2 l g 1 m 2 m 3 3 l 3 m...
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