A ball of mass $m$ is attached to the end of a thread fastened to the top of a vertical rod which is fitted to a horizontally revolving round table as shown. If the thread forms an angle $\alpha$ with the vertical, the angular velocity $\omega$ of table is

A self-propelled vehicle (assume it as a point mass) runs on a track with constant speed $v$. It passes through three positions $A, B$ and $C$ on the circular part of the track. Suppose $N_A, N_B$ and $N_C$ are the normal forces exerted by the track on the vehicle when it is passing through points $A, B$ and $C$ respectively then

Point $A$ and Point $B$ are moving in $2$ circular orbits with angular velocity $2\omega$ and $\omega$ respectively. Their positions are as shown at $t=0$. Find the time when they will meet for first time.