Three bodies, a ring, a solid cylinder and a solid sphere roll down the same inclined plane without slipping. They start from rest. The radii of the bodies are identical. Which of the bodies reaches the ground with maximum velocity?

• Ring

• Solid cylinder

• Solid sphere

• All reach the ground with same velocity

C.

Solid sphere

The total mechanical energy of a system is conserved if the forces doing work on it. 

Law of conservation of mechanical energy: the total amount of mechanical energy, in a closed system in the absence of dissipative forces ( eg. friction, air resistance), remains constant. This means that potential energy can become kinetic energy or vice versa, but energy cannot disappear. 

According to conservation of mechanical energy 

       mgh = $\frac{1}{2} \mathrm{m} {\mathrm{v}}^2 \left(1 + \frac{{\mathrm{k}}^2}{{\mathrm{R}}^2}\right)$

⇒       v² = $\left(\frac{2 \mathrm{gh}}{1 + {\displaystyle \frac{{\mathrm{k}}^2}{{\mathrm{R}}^2}}}\right)$

It is independent of the mass of the rolling body.

For a ring

          k² = R²

          v_ring = $\sqrt{\frac{2 \mathrm{gh}}{1 + 1}}$

For solid cylinder

        k² = $\sqrt{\frac{2 \mathrm{gh}}{1 + 1}}$

        k² = $\sqrt{\mathrm{g} \mathrm{h}}$

For solid cylinder

            k² = $\frac{{\mathrm{R}}^2}{2}$

     v_cylinder = $\sqrt{\frac{2 \mathrm{gh}}{1 + {\displaystyle \frac{1}{2}}}}$

                 = $\sqrt{\mathrm{g} \mathrm{h}}$

For a solid sphere

      v_sphere = $\sqrt{\frac{2 \mathrm{gh}}{1 + {\displaystyle \frac{2}{5}}}}$

        v_sphere = $\sqrt{\frac{10 \mathrm{gh}}{7}}$

Among the given three bodies the solid sphere has the greatest and the ring has the least velocity at the bottom of the inclined plane.