First slide

Rotational motion

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

Moderate

A

$\frac{M ω}{M + 4 m}$
B

$\frac{(M + 4 m) ω}{M}$
C

$\frac{(M - 4 m) ω}{M + 4 m}$
D

$\frac{M ω}{4 m}$

Solution

Initial angular momentum of ring $= I ω = M R^{2} ω$
If four objects, each of mass m, are kept gently to the opposite ends of two perpendicular diameters of the ring, then final angular momentum $= (M R^{2} + 4 m R^{2}) ω'$
By the conservation of angular momentum,
Initial angular momentum = Final angular momentum
$M R^{2} ω = (M R^{2} + 4 m R^{2}) ω' \Rightarrow ω' = (\frac{M}{M + 4 m}) ω$

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