Question

A thin circular ring of mass M and radius R is rotating about its axis with a constant angular velocity ω. Four objects each of mass m, are kept gently to the opposite ends of two perpendicular diameters of the ring. The angular velocity of the ring will be

• A. $\frac{M\omega}{M + 4m}$
• B. $\frac{(M + 4m)\omega}{M}$
• C. $\frac{(M - 4m)\omega}{M + 4m}$
• D. $\frac{M\omega}{4m}$

Answer 1

Answer: (A)

$\frac{M\omega}{M + 4m}$

Answer 2

Initial angular momentum of ring $= I\omega = MR^{2}\omega$

If four object each of mass m, and kept gently to the opposite ends of two perpendicular diameters of the ring then final angular momentum = $(MR^{2} + 4mR^{2})\omega'$

By the conservation of angular momentum

Initial angular momentum = Final angular momentum

$MR^{2}\omega = (MR^{2} + 4mR^{2})\omega'$ ⇒ $\omega' = \left( \frac{M}{M + 4m} \right)\omega$.