A thin circular ring of mass M and radius R is rotating about its axis with a constant angular velocity $\omega$ . Two objects each of mass m are attached gently to the ring. The wheel now rotates with an angular velocity

Moment of inertia of a circular ring about an axis passing through its centre and perpendicular to its plane is $M{R}^2$.

Initial angular momentum  $={\mathrm{MR}}^2\omega$

After the masses have been attached, the moment  of inertia $={\mathrm{MR}}^2+2{\mathrm{mR}}^2$

If $\omega \text{'}$   is the new angular velocity, the angular  mmentum $=({\mathrm{MR}}^2+2{\mathrm{mR}}^2){\omega }^{\mathrm{\prime }}$

By the principle of conservation of angular momentum

$({\mathrm{MR}}^2+2{\mathrm{mR}}^2){\omega }^{\mathrm{\prime }}={\mathrm{MR}}^2\omega$

$\therefore {\omega }^{\mathrm{\prime }}=\frac{\omega M}{M+2m}$