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p605hw5f08wsoln

p605hw5f08wsoln - a(1 cos θ under the action of a force...

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Phys 605. Homework 5 Due 5pm, Tuesday, October 14, 2008 Problem 5-1: [20 pts.] A spherical pendulum consists of a particle of mass m in a uniform gravitational field constrained to move on the surface of a sphere of radius R . (a) Find a Lagrangian for the spherical pendulum Find any constants of the motion. (b) Show the general motion reduces to a one-dimensional problem for which an effective po- tential can be defined. (c) Make a rough sketch of the effective potential and discuss the possible motion. (d) For proper initial conditions, the mass simply moves in a horizontal circular orbit with θ = θ 0 and constant angular frequency Ω 0 . Show that the angular frequency for small perturbations of the circular orbit is given by ω 2 = Ω 0 2 (1 + 3 cos 2 θ 0 ) . (e) Given the expression for ω in part (d) and assuming ˙ φ Ω 0 , do the orbits close in general? Briefly provide your reasoning supporting your answer. Problem 5-2: [10 pts] A particle of mass m follows the orbit

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Unformatted text preview: a (1 + cos θ ) under the action of a force always directed toward the origin. (a) Find the force law. (b) Determine the total energy of the particle in the orbit. Problem 5-3: [10 pts.] Goldstein, Poole, and Safko, (3rd. ed.) Problem 10, Chapter 3, pg. 128. The following problem was delayed until HW 6. Problem 5-4: [10 pts.] Consider point particles scattering elastically (angle of incidence equals angle of reﬂection at point of impact) from an inﬁnitely massive, perfectly hard ellipsoid of rotation. The ellipsoid is obtained by rotating the ellipse ( x 2 /a 2 ) + ( y 2 /b 2 ) = 1 about the x axis. The beam of incident particles is directed along the x-axis. (a) Show that the scattered angle Θ is related to the impact parameter s by s = b 2 a ± b 2 a 2 + tan 2 ( Θ 2 ) ²-1 / 2 (b) Find the diﬀerential scattering cross-section. (c) What is the total scattering cross-section?...
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