Question

A ring (R), a disc (D), a solid sphere (S) and a hollow sphere with thin walls (H), all having the same mass and radii, start together from rest at the top of an inclined plane and roll down without slipping.

• A. All of them will reach the bottom of the incline together.
• B. The body with the maximum radius will reach the bottom first.
• C. They will reach the bottom in order S, D, H, R.
• D. All of them will have the same kinetic energy at the bottom of the incline.

Answer 1

Answer: (C), (D)

The correct options are
C They will reach the bottom in order S, D, H, R.
D All of them will have the same kinetic energy at the bottom of the incline.

The moment of inertia of ring, disc, solid sphere and hollow sphere are $m{r}^2,\frac{1}{2}m{r}^2,\frac{2}{5}m{r}^{2}and\frac{2}{3}m{r}^2$ respectively. As the moment of inertia is the resistance to rotational motion we see that the object with least moment of inertia will reach the bottom first. Thus the order of reaching the bottom is $S,D,HandR.$

The gain in total kinetic energy is equal to loss in gravitational potential energy which in turn in same for all 4 objects and equal to $mgh$, where $h$ is height of inclined plane.