A kid of mass M stands at the edge of a platform of radius R ,...

By the principle of conservation of angular momentum about the axis of rotation of the platform, the initial angular momentum of the system is equal to the final angular momentum.

The initial angular momentum of the system is entirely due to the ball, as the platform and the kid are initially at rest. Since the ball is moving horizontally along the tangent to the edge of the platform of radius $R$, its angular momentum is:

$L_i = mvR$

After the kid catches the ball, the platform, the kid, and the ball rotate together with an angular speed $\omega$. The total moment of inertia of the system about the axis of rotation is the sum of the moments of inertia of the platform, the kid, and the ball. Both the kid and the ball are at a distance $R$ from the axis.

$I_{total} = I + MR^2 + mR^2 = I + (M + m)R^2$

The final angular momentum of the system is:

$L_f = I_{total}\omega = [I + (M + m)R^2]\omega$

Equating the initial and final angular momentum:

$mvR = [I + (M + m)R^2]\omega$

$\omega = \dfrac{mvR}{I + (M + m)R^2}$