A child is standing at one end of a long trolley moving with a speed $v$ on a smooth horizontal floor. If the child starts running towards the other end of the trolley with a speed $u$, the centre of mass of the system (trolley + child) will move with a speed.

Radius of gyration of a uniform thin rod of length $L$ about an axis passing normally through its centre of mass is

If the density of material of a circular plate and a square plate as shown is same, the centre of mass of the composite system will be 

In a bicycle the radius of rear wheel is twice the radius of front wheel. If $r_F$ and $r_r$ are the radius, $v_F$ and $v_r$ are the speeds of top most points of wheel, then

An ant is sitting at the edge of a rotating disc. If the ant reaches the other end, after moving along the diameter, the angular velocity of the disc will

A cylinder of mass $M$ and radius $r$ is mounted on a frictionless axle over a well. A rope of negligible mass is wrapped around the solid cylinder and a bucket of mass $m$ is suspended from the rope. The linear acceleration of the bucket will be