A uniform spherical shell of mass M 50 kg and radius R 130

A uniform spherical shell of mass M = 5.0 kg and radius R = 13.0 cm rotates 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 = 3.78x10^-3 kg m^2 and radius r = 6.0 cm, and its attached to a small object of mass m = 1.0 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 a distance h = 1.3 m from rest: Use work - energy considerations.

Solution

From Work enegy theorem

(1/2)I_sphereW_sphere²+(1/2)IW_pulley²+(1/2)mV²=mgh

Since Moment of inertia of sphere

I=(2/3)MR²

and W=V/r

=>(1/2)(2MR²/3)(V/R)²+(1/2)I_pulley(V/r)²+(1/2)mV²=mgh

2mgh =(2/3)MV²+I_pulley(V/R)²+mV²

2*1*9.8*1.3 =(2/3)*5*V²+(3.78*10^-3)V²/0.06²+1*V²

5.38V²=25.48

V=2.18 m/s