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

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) (M ω/4m) (B) (M ω/M+4m) (C) ((M +4m)ω/M) (D) ((M -4m)ω/M+4m)

Tardigrade Admin · Accepted Answer

According to conservation of angular momentum, L=I ω= constant. Therefore, I2 ω2=I1 ω1 or ω2=(I1 ω1/I2)=(M k2 ω/(M+4 m) k2)=(M ω/M+4 m).

Q. 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

$\frac{M ω}{4 m}$

$\frac{M ω}{M + 4 m}$

$\frac{( M + 4 m ) ω}{M}$

$\frac{( M - 4 m ) ω}{M + 4 m}$

According to conservation of angular momentum,
$L = I ω =$ constant.
Therefore, $I_{2} ω_{2} = I_{1} ω_{1}$
or $ω_{2} = \frac{I _{1} ω _{1}}{I _{2}} = \frac{M k ^{2} ω}{( M + 4 m ) k ^{2}} = \frac{M ω}{M + 4 m}$ .

Q. A thin circular ring of mass Mand 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

4mMω​

M+4mMω​

M(M+4m)ω​

M+4m(M−4m)ω​