A uniform rod of mass m and length l is struck at an end by a force F perpendicular to the rod for a short time interval. Calculate

(a) the speed of the center of mass, (b) the angular speed of the rod about the center of mass, (c) the kinetic energy of the rod and (d) the angular momentum of the rod about the center of mass after the force has stopped to act. Assume that t is so small that the rod does not appreciably change its direction while the force acts.

Suppose the platform of the previous problem is brought to rest with the bail in the hand of the kid standing on the rim. The kid throws the ball horizontally to his friend in a direction tangential to the rim with a speed u as seen by his friend. Find the angular velocity with which the platform will start rotating.

Suppose the platform with the kid in the previous problem is rotating in anticlockwise direction at an angular speed w. The kid starts walking along the rim with a speed u relative to the platform also in the anticlockwise direction. Find the new angular speed of the platform.

A uniform rod of length L lies on a smooth horizontal table. A particle moving on the table strikes the rod perpendicularly at an end and stops. Find the distance travelled by the center of the rod by the time it turns through a right angle. Show that if the mass of the rod is four times that of the particle, the collision is elastic.

Suppose the particle of the previous problem has a mass m and a speed u before the collision and it sticks to the rod after the collision. The rod has a mass M. (a) Find the velocity of the center of mass C of the system constituting “the rod plus the particle”. (b) Find the velocity of the particle with respect to C before the collision. (c) Find the velocity of the rod with respect to C before the collision. (d) Find the angular momentum of the particle and of the rod about the center of mass C before the collision. (e) Find the moment of inertia of the system about the vertical axis through the center of mass C after the collision. (f) Find the velocity of the center of mass C and the angular velocity of the system about the center of mass after the collision.