Question

Consider a situation in which a ring, a solid cylinder and a solid sphere roll down on the same inclined plane without slipping. Assume that they start rolling from rest and having identical diameter. The correct statement for this situation is:

• A. All of them will have same velocity.
• B. The cylinder has the greatest and the sphere has the least velocity of the centre of mass at the bottom of the inclined plane.
• C. The ring has the greatest and the cylinder has the least velocity of the centre of mass at the bottom of the inclined plane.
• D. The sphere has the greatest and the ring has the least velocity of the centre of mass at the bottom of the inclined plane.

Answer 1

The correct option is D The sphere has the greatest and the ring has the least velocity of the centre of mass at the bottom of the inclined plane.

We know that, the velocity of the centre of mass, at the bottom, of a body rolling in an inclined plane of height $h$ is given by,

$v=\frac{\sqrt{2gh}}{\sqrt{1+\frac{I}{m{R}^2}}}$

$\therefore v\propto \frac{1}{\sqrt{I}}$

$(I_{\text{ring}}=M{R}^2)>(I_{\text{solid cylinder}}=\frac{1}{2}M{R}^2)>(I_{\text{solid sphere}}=\frac{2}{5}M{R}^2)$

$\Rightarrow v_{\text{ring}}<v_{\text{solid cylinder}}<v_{\text{solid sphere}}$

Hence, $\left(A\right)$ is the correct answer.