Question

A hollow sphere and a solid sphere having equal mass and equal radii are rolled down without slipping on a rough inclined plane.

• A. The two spheres reach the bottom simultaneously
• B. The hollow sphere reaches the bottom with lesser speed
• C. The solid sphere reaches the bottom with greater kinetic energy
• D. The two spheres will reach the bottom with same linear momentum

Answer 1

The correct option is C The solid sphere reaches the bottom with greater kinetic energy

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 more in hollow sphere as in hollow sphere , all mass is concentrated away than in solid sphere so it has more moment of inertia and hence more radius of gyration.

So solid sphere will reach the bottom earlier.