A thin circular ring of mass M and radius R rotates about an axis through its centre and perpendicular to its plane, with a constant angular velocity ω. Four small spheres each of mass m (negligible radius) are kept gently to the opposite ends of two mutually perpendicular diameters of the ring. The new angular velocity of the ring will be

Question

A thin circular ring of mass M and radius R rotates about an axis through its centre and perpendicular to its plane, with a constant angular velocity ω. Four small spheres each of mass m (negligible radius) are kept gently to the opposite ends of two mutually perpendicular diameters of the ring. The new angular velocity of the ring will be (A)  4ω  (B)  (M/4m)ω  (C)  ( (M+4m/M) )ω  (D)  ( (M/M-4m) )ω

Tardigrade Admin · Accepted Answer

According to conservation of angular momentum, Iω =constant ie, we can write I1ω 1=I2ω 2 or MR2ω =(M+4m)R2ω 2 or ω 2=( (M/M+4m) )ω

$4 ω$

$\frac{M}{4 m} ω$

$(\frac{M + 4 m}{M}) ω$

$(\frac{M}{M - 4 m}) ω$

$(\frac{M}{M + 4 m}) ω$

According to conservation of angular momentum,
$I ω = co n s t an t$ ie, we can write $I_{1} ω_{1} = I_{2} ω_{2}$
or $M R^{2} ω = (M + 4 m) R^{2} ω_{2}$
or $ω_{2} = (\frac{M}{M + 4 m}) ω$

4ω

4mM​ω

(MM+4m​)ω

(M−4mM​)ω

(M+4mM​)ω