A thin circular ring of mass $M$ and radius $R$ is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity $\omega$. If two objects each of mass $m$ be attached gently to the opposite ends of a diameter of the ring, the ring will then rotate with an angular velocity:

Since no external torque acts on the system, the total angular momentum $L$ is conserved ($L_i=L_f$).Initial moment of inertia: $I_i=M{R}^2$ Initial angular velocity: ${\omega }_i=\omega$ Initial angular momentum: $L_i=I_i\omega =M{R}^2\omega$ When two masses $m$ are attached at the ends of the diameter (distance $R$ from the axis):4. Final moment of inertia: $I_f=M{R}^2+m{R}^2+m{R}^2=(M+2m)R^2$ 5. Final angular velocity: ${\omega }_f$ Using conservation of angular momentum:
$$I_i{\omega }_i=I_f{\omega }_f$$
$$M{R}^2\omega =(M+2m)R^2{\omega }_f$$
$${\omega }_f=\frac{M\omega }{M+2m}$$