Question

a uniform spherical shell of mass M and radius R rotates about a vertical axis on frictionless bearing. A massless cord passes around the equator of the shell, over a pulley of rotational inertia I and radius r, and is attached to a small object of mass m. 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 from rest? use work-energy considerations.

Answer 1

since no information is given about the initial velocity of any object we assume the whole system started from rest.

Now,

Given

mass of the shell = M

radius of the shell = R

Moment of inertia of the pulley = I

radius of the pulley = r

now after falling a distance h

velocity of the block be v

angular velocity of the shell be

So, velocity of the shell =

angular velocity of the pulley be

So, velocity of the pulley =

Now, since the rope has a fixed length and it is tension so if any point is moving in a certain speed other points on the rope must have the same speed.

So, Speed of rope at shell=speed of rope at the pulley= speed of the point attached to mass m

................(a)

Now, by conservation of energy

U_i + K_i=U_f+K_{f...........................(i)}

let the potential energy of the object at the start be 0

So, U_i = 0 as both pulley and shell change potential energy

so, potential energy after falling height h = -mgh

U_f = -mgh

Now,

K_i=0 since the system start from rest

and using equation (a)

So, equation (i) becomes

which is the required solution