A ring, a solid sphere and a thin disc of different masses rotate with the same kinetic energy. Equal torques are applied to stop them. Which will make the least number of rotations before coming to rest

To determine which object makes the least number of rotations before coming to rest, let us analyze the relationship between rotational kinetic energy, torque, and angular displacement (or rotations).

The rotational kinetic energy ($K_R$) of a rotating body is given by the work-energy theorem for rotational motion. When a retarding torque ($\tau$) is applied to bring a rotating body to rest, the work done by the torque is equal to the change in its rotational kinetic energy.

Mathematical expression for the work-energy theorem in rotational motion:
$W=\tau ·\theta =\Delta K_R$
where:
- $\tau$ is the applied torque,
- $\theta$ is the total angular displacement (in radians) before coming to rest, and
- $\Delta K_R$ is the change in rotational kinetic energy.

Since all the bodies (the ring, the solid sphere, and the thin disc) are brought to rest, their final kinetic energy is zero. Thus, the change in kinetic energy is equal to their initial kinetic energy:
$\Delta K_R=K_R$

This gives the relation:
$\tau ·\theta =K_R$

Rearranging the equation to solve for the angular displacement ($\theta$):
$\theta =\frac{K_R}{\tau }$

The number of rotations ($N$) made by a body is directly proportional to its angular displacement ($\theta$), defined by:
$N=\frac{\theta }{2\pi }=\frac{K_R}{2\pi ·\tau }$

From the problem statement, we are given that:
1. All three bodies rotate with the same initial kinetic energy ($K_R$ is constant).
2. Equal torques are applied to stop them ($\tau$ is constant).

Since both $K_R$ and $\tau$ are the same for the ring, the solid sphere, and the thin disc, their angular displacement $\theta$ (and consequently, the number of rotations $N$) must also be identical, regardless of their masses, shapes, or moments of inertia.

Therefore, all will make the same number of rotations before coming to rest.