physics 1 moment of inertia.

Question

An object rolls without slipping down a 2.03 m high incline. What is the moment of inertia of the object starting from rest if it has a final velocity of 6.1 m/s? Express the moment of inertia as a multiple of MR² , where M is the mass of the object and R is its radius. 

Answer 1

When a body rolls down without slipping from an inclined plane

At the bottom of incline it has two energies KE of translation and KE of rotation

so

Mgh= = 1/2 Mv² +1/2 Iw²

Where I = MOI, v = Velocity, w= v/R, w= Angular velocity

w=v/R , where R = Radius

So 1/2Iw² = Mgh -1/2 Mv²

1/2 ( I ) (v/R) ² = Mgh -1/2 Mv²

Plug in g= 9.8, v=6.1 , h= 2.03

you get I in terms of M and R