A uniform spherical shell of mass M = 5.0 kg and radius R = 8.6 cm can rotate ab

Question

A uniform spherical shell of mass M = 5.0 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 10-3 kg · m2 and radius r = 5.0 cm, and is attached to a small object of mass m = 0.60. 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 93 cm after being released from rest? Use energy considerations.

Explanation / Answer

Applying law of conservation of energy we ve

mgh = 1/2 mv^2 + 1/2 Iw^2 + 1/2 Iw^2

MI of sphere = 2/3 Mr^2

and w = v/r

hence the above equation becomes

mgh = 1/2 mv^2 + 1/2 (2/3Mr^2)(v/r)^2 + 1/2 I(v/r)^2

which on simplifying gives

v = sqrt(mgh/(M/3+ m/2 + I/2r^2))

v = sqrt(0.6*9.8*0.93)/(5/3 + 0.6/2 + 3*10^-3/(2*25*10^-4)) = 1.4595 m/sec

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