A uniform spherical shell of mass M = 17.0 kg and radius R = 0.710 m can rotate

Question

A uniform spherical shell of mass M = 17.0 kg and radius R = 0.710 m can rotate about a vertical axis on frictionless bearings (see the figure). A massless cord passes around the equator of the shell, over a pulley of rotational inertia I = 0.200 kg·m2 and radius r = 0.130 m, and is attached to a small object of mass m = 2.10 kg. There is no friction on the pulley's axle; the cord does not slip on the pulley. What is the speed of the object when it has fallen a distance 0.812 m after being released from rest? Use energy considerations.

Explanation / Answer

applying energy conservation,

PEi + KEi = PEf + KEf

0 + 0 = - m g h + (m v^2 /2 ) + (I w^2 / 2) + ( I' w'^2 / 2)

2 m g h = m v^2 / 2 + I (v/r)^2 + (2 M R^2 / 3) (v/R)^2

2 x 2.10 x 9.81 x 0.812 = 2.10 v^2 + (0.200/0.130^2)v^2 + (2 x 17/3) v^2

33.456 = 25.27 v^2

v = 1.15 m/s

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