A uniform spherical shell of mass M = 16.0 kg and radius R = 0.500 m can rotate

Question

A uniform spherical shell of mass M = 16.0 kg and radius R = 0.500 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 = 0.210 kg ·m2 and radius r = 0.0750 m, and is attached to a small object of mass m = 4.20 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 1.38 m after being released from rest? Use energy considerations.

Explanation / Answer

applying law of conservation of energy,

0.5*Iawa^2 +0.5*Ibwb^2+0.5mv^2 = mgh

but wa*ra=wb*rb=v

also,

Ib=(2/3)Mr^2

so,

0.5*(2/3)*16*v^2 +0.5*4.2*v^2+ (0.5*0.210 *v^2/ 0.075^2) =4.2*9.8*1.38

or v=1.475 m/s

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