Question

A solid sphere is rolling down on inclined plane from rest and rectangular block of same mass is also slipping down simultaneously from rest on a similar smooth inclined plane.

• A. Both of them will reach the bottom simultaneously.
• B. The sphere will reach the bottom first.
• C. The rectangular block will reach the bottom first.
• D. Depends on density of material

Answer 1

The correct option is B The sphere will reach the bottom first.

In rolling without slipping, no work is done against friction. Hence, loss in gravitational potential energy of a body is equal to its kinetic energy,i.e., linear plus rotational kinetic energies.

Also, $v=mr$

Total kinetic energy

$=\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2⟹K{E}_{total}=\frac{1}{2}m{v}^2[1+\frac{K^2}{r^2}]$

where $K$ is radius of gyration.

As all the bodies have same final total kinetic energy, their final velocities will depend on the ratio $\frac{K}{r}$.

Bodies with smaller value of $\frac{K}{r}$ will have greater velocity and hence will reach the bottom earlier.

And the ratio of $\frac{K}{r}$ is less for solid sphere.and hence less moment of inertia and less radius of gyration.

So solid sphere will reach the bottom earlier.