$A$ block of mass $m$ is attached to a pulley disc of equal mass $m$ and radius $r$ by means of a slack string as shown. The pulley is hinged about its centre on a horizontal table and the block is projected with an initial velocity of $5\, m/s$. Its velocity when the string becomes taut will be

$A$ roller is made by joining together two cones at their vertices $O$. It is kept on two rails $AB$ and $CD$,which are placed asymmetrically (see figure),with its axis perpendicular to $CD$ and its centre $O$ at the centre of the line joining $AB$ and $CD$ (see figure). It is given a light push so that it starts rolling with its centre $O$ moving parallel to $CD$ in the direction shown. As it moves,the roller will tend to

An object has a moment of inertia of $3 \ kg \cdot m^2$. It is rotating with an angular velocity of $2 \ rad/s$. If a mass of $12 \ kg$ is moving with a velocity of $v \ m/s$,at what value of $v$ will their kinetic energies be equal?

$A$ disc and a ring of the same mass are rolling. If their kinetic energies are equal,then the ratio of their velocities will be:

$A$ hinged construction consists of three rhombs with the side ratio $5:3:2$. Vertex $A_3$ moves in the horizontal direction at a velocity $v$. The velocity of $A_2$ is ....... $V$.

One ice skater of mass $m$ moves with speed $2v$ to the right,while another of the same mass $m$ moves with speed $v$ toward the left,as shown in figure $I$. Their paths are separated by a distance $b$. At $t = 0$,when they are both at $x = 0$,they grasp a pole of length $b$ and negligible mass. For $t > 0$,consider the system as a rigid body of two masses $m$ separated by distance $b$,as shown in figure $II$. Which of the following is the correct formula for the motion after $t = 0$ of the skater initially at $y = b/2$?