A uniform spherical shell of mass M = 3.6 kg and radius R = 8.6 cm can rotate ab

Question

A uniform spherical shell of mass M = 3.6 kg and radius R = 8.6 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 = 3.0 103 kg  · m2 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 85 cm after being released from rest? Use energy considerations.
m/s

Explanation / Answer

Using Conservation of energy

Potential energy of Block = Kinetic energy of Block + Rotational KInetic energy of pulley + Rotational KE of sphere

mgh = (0.5) m v2 + (0.5) Ip (v/r)2 + (0.5) Is (v/R)2

mgh = (0.5) m v2 + (0.5) (0.003) (v/r)2 + (0.5) (2/3) (MR2) (v/R)2

(0.6) (9.8) (0.85) = (0.5) (0.6) v2 + (0.5) (0.003) (v/0.05)2 + (0.5) (2/3) ((3.6) (0.086)2) (v/(0.086))2

v = 1.54 m/s

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