A uniform spherical shell of mass M = 18.0 kg and radius R = 0.600 m can rotate

Question

A uniform spherical shell of mass M = 18.0 kg and radius R = 0.600 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 = 0.0870 kg·m2 and radius r = 0.140 m, and is attached to a small object of mass m = 2.80 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 0.925 m after being released from rest? Use energy considerations.

Explanation / Answer

Here,

for the spherical shell ,

moment of inertia , I1 = 0.4 * M * R^2

I1 = 0.5 * 18 * 0.60^2

I1 = 3.24 Kg.m^2

let the speed of object is v

Using conservation of energy

0.5 * I1 * w1^2 + 0.5 * I * w2^2 + 0.5 * m * v^2 = m * g * h

0.5 * 3.24 * (v/0.60)^2 + 0.5 * 0.0870 * (v/0.140)^2 + 0.5 * 2.80 * v^2 = 2.80 * 9.8 * 0.925

solving for v

v = 1.77 m/s

the speed of the block after falling 0.925 m is 1.77 m/s

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