Question

A uniform disc of mass m and radius R is hanging from a rigid support and is free to rotate about a horizontal axis passing through its center in vertical plane as shown. An insect of mass m falls vertically and hits the disc at a point at horizontal diameter with a velocity $v_0$ and sticks on it. calculate the minimum value of $v_0$ so that the disc completes the vertical circular motion.

Answer 1

Conserving angular momentum of the whole system about the CM of the disc
$m{v}_{0}R=(\frac{M{R}^2}{2}+m{R}^2)\omega$

$\Rightarrow \omega =\frac{2m{v}_0}{(M+2m)R}$ ............(I)

From conservation of energy thereafter ,
$mgR=\frac{1}{2}(\frac{M{R}^2}{2}+m{R}^2){\omega }^2$..........(ii)

Putting values of $\omega$ from Eq. (i) in Eq.(ii) we get,
$v_0=\sqrt{\frac{(M+2m)gR}{m}}$