A uniform spherical shell of mass M = 16.0 kg and radius R = 0.450 m can rotate

Question

A uniform spherical shell of mass M = 16.0 kg and radius R = 0.450 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 = 0.110 kg-m^2 and radius r = 0.100 m, and is attached to a small object of mass m = 3.30 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 0.784 m after being released from rest Use energy considerations.

Explanation / Answer

After falling a distance h, Potential energy m g h is converted to kinetic energy of the rotating shell, the falling mass, and the pulley. Let V be the velcoity of the mass at time t.
The angular velocity of the shell is ws= V/R and its moment of inertia is
Is = (2/3)MR^2
The moment of inertia of the pulley is I, its angular velocity is wp = V/r, and its radius is r.

mgh = (1/2)mV^2 + (1/2)(2/3)MR^2*(V/R)^2 + (1/2) I (V/r)^2
= V^2 [(1/2)m + (1/3)M + (1/2)I/r^2]
V^2 = 2gh/[1 + (2/3)M/m + I/(mr^2)]

v^2 = 15.36/(1+3.23+3.33) = 15.36/7.56

v = 1.425 m/sec

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