Question

A thin circular ring of mass m and radius r is rotating

Answer 1

Correct question :

A thin circular ring of mass M and radius r is rotating about its axis with a constant angular velocity ω. Two objects, each of mass m, are attached gently to the opposite ends of the diameter of the ring. The wheel now rotates with an angular velocity

Answer :

Let the final angular velocity be w'

Angular momentum is conserved about center of the ring.

$\sf L_i=L_f$

$\bf iw + 0 = iw' + 2 \times m{r}^{2} w' \\ \\ \bf M{r}^{2} w + 0 = M{r}^{2} w' + 2 \times m {r}^{2} w' \\ \\ \boxed{\bf w' = \frac{Mw}{M + 2m}}$