A hollow and a solid cylinder, with the same mass and radius, roll down an incli

Question

A hollow and a solid cylinder, with the same mass and radius, roll down an incline without dissipation of mechanical energy, both starting from rest at the same height. (See the figure.) The __1__ (choose one of hollow cylinder|solid cylinder) has the greater moment of inertia. When they reach the bottom, the kinetic energy of the hollow cylinder is __2__ (greater than|equal to|less than) that of the solid cylinder. The __3__ (hollow cylinder|solid cylinder) has a higher fraction of its total kinetic energy in the form of rotational kinetic energy. At the bottom of the incline, the __4__ (hollow cylinder|solid cylinder) has the greater speed.

Explanation / Answer

The moment of inertia of hollow cylinder = MR^2 while that of the solid cylinder = MR^2/2

1=> Hollow cylinder has greater moment of inertia.

the work done on both cylinders are same as the height initially is same.

So the change in kinetic energy must be same .

2=> Both have equal kinetic energy.

The ratio of rotational kinetic energy and translational kinetic energy = 0.5 I omega^2/(0.5 mv^2) = 0.5kMR^2omega^2/(0.5*Momega^2R^2) = k.

3=> Hallow cylinder have higher fraction of rotational kinetic energy.

k times the translational kinetic energy is same for both.

Hence the ratio of velocities will be opposite of the k value.

4=> Solid cylinder will have greater speed at the bottom.

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