2002AugQualMECH - = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ − =...

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Ben Sauerwine Practice for Qualifying Exams Problem Source: CMU Qualification Exam Day 2 (August 2002) (2) Consider a solid uniform sphere of radius a and mass m which rolls without slipping inside a fixed cylindrical pipe of radius R. Neglect friction, and let the sphere move only in the plane shown by the figure: (a) Show that the moment of inertia of a uniform sphere is given by 2 5 2 ma I = .
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() () () 2 0 3 0 2 0 3 0 3 5 3 0 2 00 3 4 3 0 2 00 2 2 3 5 2 3 4 3 2 2 cos 3 1 2 sin cos 1 sin sin 2 5 4 3 sin 3 4 sin sin 3 4 ma d d use d a a m dr d d r a m dr d d r r a m a a →= = = + = = →= →= ∫∫∫ ∫∫∫ π ππ φ φφφ φφ φφπ θφφ (b) Find the equation of motion for the sphere. Now, taking the angle theta to be the angle between the center of the ball and the vertical, I have: () () ( ) () θ θθ sin 1 cos 1 1 2 1 2 1 2 1 2 2 2 2 2 2 2 2 a R mg mR I a R I L L dt d V K L a R mg V mR I a R I mR I a R I K = + +
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Unformatted text preview: = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ − = − − = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = & & & & & & & (c) Find the period of small oscillations about the equilibrium position. Expanding the sine term, I have ( ) a R mg mR I a R I − ≈ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − & & 2 2 1 This has solutions: ( ) ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = t mR I a R I a R mg A 2 2 1 cos θ and so the period for small oscillations is: ( ) a R mg mR I a R I T − + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = 2 2 1 2 π...
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This note was uploaded on 04/07/2012 for the course PHYSICS 767 taught by Professor Dr.jaouni during the Spring '12 term at Abu Dhabi University.

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2002AugQualMECH - = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ − =...

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