University Physics with Modern Physics with Mastering Physics (11th Edition)

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9.84: Taking the zero of gravitational potential energy to be at the axle, the initial potential energy is zero (the rope is wrapped in a circle with center on the axle).When the rope has unwound, its center of mass is a distance R π below the axle, since the length of the rope is R 2 and half this distance is the position of the center of the mass. Initially, every part of the rope is moving with speed , 0 R ϖ and when the rope has unwound, and the cylinder has angular speed
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Unformatted text preview: , the speed of the rope is R (the upper end of the rope has the same tangential speed at the edge of the cylinder). From conservation of energy, using 2 ) 2 1 ( MR I = for a uniform cylinder, . 2 4 2 4 2 2 2 2 R mgπ R m M R m M-ϖ + = ϖ + Solving for gives ( 29 ( 29 , 2 4 2 m M R πmg ω ω + + = and the speed of any part of the rope is . R v =...
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