As shown in the figure,a small wheel is coaxially connected to a larger wheel of double its radius. The system rotates about a common axis. Strings $A$ and $B$ are attached to the wheels such that they do not slip. If $x$ and $y$ are the distances covered by $A$ and $B$ in the same time interval,then:

$A$ thin rod is hinged at point $O$ and is in an unstable equilibrium position. It falls under the influence of gravity due to a slight disturbance. It makes angles of $60^{\circ}$,$90^{\circ}$,and $180^{\circ}$ with the vertical in positions $(2)$,$(3)$,and $(4)$ respectively. If $\omega_2$,$\omega_3$,and $\omega_4$ are the angular velocities in these positions,then:

The instantaneous velocity of point $B$ of a rod of length $L = 0.5 \ m$ is $v = 3 \ m/s$ at an angle of $30^{\circ}$ with the rod as shown. The angular velocity of the rod for the minimum velocity of end $A$ is:

$A$ uniform solid cylinder of mass $M$ and radius $R$ rotates about a frictionless horizontal axle. Two similar masses $m$ are suspended with the help of two ropes wrapped around the cylinder. If the masses fall down through a distance $h$,the angular velocity of the cylinder will be

$A$ ball of mass $m$ is released from inside a smooth wedge of mass $m$ as shown in the figure. What is the speed of the wedge when the ball reaches point $B$?

$A$ mass $m$ is supported by a massless string wound around a uniform hollow cylinder of mass $m$ and radius $R$. If the string does not slip on the cylinder, then with what acceleration will the mass be released? (Assume $g =$ acceleration due to gravity)