A uniform spherical shell of mass M = 17.0 kg and radius R = 0.410 m can rotate

Question

A uniform spherical shell of mass M = 17.0 kg and radius R = 0.410 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 middot m^2 and radius r = 0.130 m, and is attached to a small object of mass m = 4.90 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 1.19 m after being released from rest? Use energy considerations. Number ______ Units m/s

Explanation / Answer

Using energy conservation:

KEi + PEi = KEf + PEf

0 + m*g*d = 0.5*m*v^2 + 0.5*Ip*wp^2 + 0.5*Is*ws^2 + 0

Ip = moment of inertia of pulley = 0.2 kg-m^2

Is = moment of inertia of sphere = 2MR^2/5 = 2*17*0.41^2/5 = 1.143 kg-m^2

v = ws*R = wp*r

ws = v/R, wp = v/r

d = 1.19 m, R = 0.410 m, r = 0.13 m

4.9*9.81*1.19 = 0.5*4.9*v^2 + 0.5*1.143*v^2/0.41^2 + 0.5*0.2*v^2/0.13^2

57.202 = v^2*(2.45 + 3.399 + 5.917)

v = sqrt (57.202/(2.45 + 3.399 + 5.917))

v = 2.204 m/sec

Let me know if you have any doubt.

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