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A uniform spherical shell of mass M and radius R rotates about a vertical axis on frictionless bearing. A massless cord passes around the equator of the shell, over a pulley of rotational inertia I and radius R and is attached to a small object of mass m that is otherwise free to fall under the influence of gravity. There is no friction of pulley's axle; the cord does not slip on the pulley. What is the speed of the object after it has fallen a distance h from rest? Use work-energy considerations.

Solution
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By conservation of energy, when object falls by a distance of h.12mv2+12Ishellω2shell+12Ipulley=mghv=Rω12mv2+12(23MR2)×(vR)2+12Ip(vrp)2=mghv2(m+23M+Ipr2p)=2mghv=   ⎷2mgh(m+23M+Ipr2p)

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