A uniform spherical shell of mass M = 4.0 kg and radius R = 9.1 cm can rotate ab

Question

A uniform spherical shell of mass M = 4.0 kg and radius R = 9.1 cm can rotate about a vertical axis on frictionless bearings (see figure below). A massless cord passes around the equator of the shell, over a pulley of rotational inertia I = 3.0 Times 10^-3 kg m^2 and radius r = 5.0 cm, and is attached to a small object of mass m = 0.60 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 after it has fallen 92 cm after being released from rest? Use energy considerations.

Explanation / Answer

Gain in kinetic energy of mass + rotational KE pulley + rotational KE sphere = loss in potential energy of mass

Moment of inertia of sphere = 2/5 * 4 * 0.091 * 0.091

                                             =   0.01325 kg-m2

=> 0.5 * 0.60 * v2 + 0.5 * 3*10-3 * v2/(2.5*10-3) + 0.5 * 0.01325 * v2/(8.281*10-3) =   0.60 * 9.8 * 0.92

=>   0.3v2 + 0.6v2 + 0.8v2 = 5.41

=>   1.7v2 = 5.41

=>   v = 1.784 m/sec      --------------> speed of object

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