Suppose in the previous poroblem, the child stays at rest on the platform and one of hils firend throws a ball of mas `m_(1)` towards thim with a velocity of `v` from a direction tangential to the platform and the child on the platform catches the ball. Find the angular velocity of the platform after he catches the ball.

Explanation

Again in this case no external torque is acting on the system so we can equate the angular momentum before catch and after catch. If after catching the ball, platform starts rotating with an angular speed `omega`, we have
Angular momentum of the ball about centre of platform before catch `=` Angular momentum of the platform plus child plus ball after catch
`m_(1)uLR=(I+mR^(2)+m_(1)R^(2))omega`
or `omega=(m_(1)uR)/(I+mR^(2)+m_(1)R^(2))`
Here we can conserve angular momentum with respect to the centre pivot of the platform. This is because when the child catches the ball, it has a linear momentum and due to it the plat-form tends to gain a linear momentum, which develops a normal reaction on the pivot in opposite direction and it will not allow the platform to move translationally. If the platform was resting on a smooth plane without pivot at the centre, after catching the ball, the platform would also move translationally with its rotational motion and in that case linear momentum of system will also remain conserved.