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aging Qualiﬁcation Examination for PhD candidates ( Oct 2007 ) Classical Mechanics A uniform rod with mass M and radius it rolls without slide in a bigger cylinder with radius R.(a) Calculate the moment of inertial through the axis for the rod (7%)
(b) Set up the equation of motion. (7%) (C) Calculate the frequency of small angle
oscillation. (7%) . A vibration system is composed of two masses 9m and m attached to the wall and rests on a smooth horizontal table, as shown in the ﬁgure. (a) Find the eigen—
frequencies of the system. (7%) (b) Find the normalized eigen~vectors. (7%) rest positions /\ I 1 9m m . Apply the Hamilton—Jacobi theory to solve the simple harmonic oscillation with H = 217—; + $22. Also, ﬁnd out the oscillation frequency by using the action—angle variable. (15%) A planet of mass M and radius R is passing through a meteor shower with uniform
distribution at a relative velocity 1}. Assuming the classical Newtonian gravitational interaction, find out the effective cross section for this poor planet to be struck by
the meteor.(15%) . Consider an 1D simple harmonic oscillation system which can be described by 1 a Lagrangian L m §mﬁ32 — Ekxg + zF(t). Determine the ﬁnal amplitude for the oscillations under a ‘force’ F(t) which is zero for t < O and t > T, and F(t) I Fort/T
for 0 < t < T, if up to time t = 0 the system is at rest in equilibrium. (15%) . A particle is thrown up vertically with initial speed 220, reaches a maximum height and falls back to ground. For any given location on the surface of the Earth, show
that the Coriolis deﬂection when it again reached the ground is opposite in direction,
and four times greater in magnitude, than the Coriolis deflection when it is dropped
at rest from the same maximum height.(20%) ...
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 Spring '09
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