Question

A uniform horizontal circular disc of mass $M$ and radius $R$ can freely rotate about a vertical axle passing through its centre. A particle of mass $m$ is placed near the centre of the disc in a smooth groove made along the radius of the disc as shown. An initial angular velocity ${\omega }_2$ is imparted to the disc. Find the angular velocity of the disc, when the particle reaches the other end of the groove. (Take $\frac{M}{m}=4$)

Answer 1

No extend force so total angular

momentum is constant

$I_1{\omega }_f=I,{\omega }_i$

${\omega }_f=\left(\frac{I_i}{I_f}\right)\omega ;$

$I_i=\frac{1}{2}M{R}^2+\frac{1}{2}M(0{)}^2=\frac{1}{2}M{R}^2(M=4x)$

$I_i=2M{R}^2$

$I_f=\frac{1}{2}M{R}^2+\frac{1}{2}M(R{)}^2=\frac{5}{2}M{R}^2$

$\frac{I_i}{I_f}=\frac{4}{5}$

${\omega }_f=\frac{4}{5}{\omega }_i$