A thin circular ring of mass M and radius R is rotating with a...

As the masses are gently placed we can apply conservation of angular momentum.

Initial moment of Inertia ${I}_{1} = M{R}^{2}$

Final moment of inertia ${I}_{2} = M{R}^{2} + \left(2m\right){R}^{2}$

By conservation of angular momentum

${I}_{1}{\omega}_{1} = {I}_{2}{\omega}_{2}$$⇒ M{R}^{2} \times 2 = \left(M + 2m\right){R}^{2} \times {\omega}_{2}$

$⇒ {\omega}_{2} = \dfrac{2M}{M + 2m}rad{s}^{- 1}$