Question

A solid sphere, a hollow sphere and a ring, all having equal mass and radius, are placed at the top of an incline and released. The friction coefficients between the objects and the incline are equal but not sufficient to allow pure rolling. The greatest kinetic energy at the bottom of the incline will be achieved by:
(A) The solid sphere
(B) The hollow sphere
(C) The ring
(D) All will achieve the same kinetic energy.

Answer 1

Hint
As we know that the kinetic energy is $KE = \dfrac{1}{2}I{\omega ^2}$, as the kinetic energy directly depends upon the moment of inertia. And we also know that the moment of inertia of the solid sphere is $I = \dfrac{{2m{r^2}}}{5}$and for the hollow sphere is $I = \dfrac{{2m{r^2}}}{3}$, the moment of inertia of the ring is $I = m{r^2}$as they all have same mass and the radius the we can put these values in the formula of kinetic energy, then we will get the required results.

Complete step by step answer
Let us consider a solid sphere, hollow sphere and a ring, all having the same mass and radius, and placed at the top of the incline and release. The friction is not sufficient for pure rolling.
Now, as we know that kinetic energy of the object which is rotating with angular velocity ω, then $KE = \dfrac{1}{2}I{\omega ^2}$
where, $I$ is the moment of inertia about the centre of mass.
Now, as kinetic energy directly depends upon the moment of inertia then we can put the values of the moment of inertia of objects.
The moment of inertia for solid sphere is $I = \dfrac{{2m{r^2}}}{5}$
then kinetic energy of the solid sphere is, $K{E_S} = \dfrac{{m{r^2}{\omega ^2}}}{5} = 0.2m{r^2}{\omega ^2}$ … (1)
now, the moment of inertia of the hollow sphere is $I = \dfrac{{2m{r^2}}}{3}$
then the kinetic energy of the hollow sphere is $K{E_h} = \dfrac{{m{r^2}{\omega ^2}}}{3} = 0.33m{r^2}{\omega ^2}$ ... (2)
Now, the moment of inertia of ring is -
Then, the kinetic energy of the ring is $K{E_r} = \dfrac{{m{r^2}{\omega ^2}}}{2} = 0.5m{r^2}{\omega ^2}$ … (3)
Hence, from equations (1), (2) and (3), we get that the Ring has the greatest kinetic energy.
Therefore, option (C) is correct.

Note
For finding out the rotational kinetic energy of the bodies, we use the concept of moment of inertia. Moment of inertia is the quantity that expresses the tendency of the body to resist angular acceleration. It is the product of mass of every particle of the body with the square of its distance from the axis of the rotation.