Answer to Question #86806 in Mechanics | Relativity for Shahi

Question

A body is rolling of mass m and radius r is rolling without slipping on a smooth horizontal surface towards smooth vertical wall. The body collides elastically with the wall and it is observed that about point in return journey , the angular momentum about a point is zero after the collision. The body can be? Briefly explain this concept please
(A) Hollow Sphere. (b) Hollow cylinder
(C) Solid Sphere (d) solid cylinder

Answer 1

Write law of conservation of energy for the rolling body. Before hitting the wall the kinetic energy was

$$KE_1=\frac{1}{2}mv^2+\frac{1}{2}I\omega^2,$$

and after the collision the energy became 0 since according to the condition the body stopped. Rewrite the firs equation:

$$KE_1=\frac{1}{2}m(\omega r)^2+\frac{1}{2}I\omega^2,$$

and since the body rolled without slipping, there is no energy lost to friction:

$$\mu mgx=0,$$

and thus we have no expressions that allow us to identify what the body looked like.

Let's suppose that body lost energy for something which caused the body to stop:

$$\frac{1}{2}m(\omega r)^2+\frac{1}{2}I\omega^2=E_\text{lost1}+\frac{1}{2}m(\omega_\text{final} r)^2+\frac{1}{2}I\omega^2_\text{final},$$

$$\frac{1}{2}m(\omega_\text{final} r)^2+\frac{1}{2}I\omega^2_\text{final}=E_\text{lost2},$$

and the body stops.

Indeed, if there is no friction between the wall and the body, any body could do that.

Let's look at the problem from another angle. It is stated that the body returned to its starting point, it is less possible for spherical objects but more possible for cylindrical ones because they can easily rebound under 90 degree angle to the wall.