The centre of a disc rolling without slipping on a plane surface moves with speed$u.$ A particle, on the lower half of the rim making an angle $60°$ with vertical, will be moving at speed 

A thin string is wrapped several times around a cylinder kept on a rough horizontal surface. A boy standing at a distance $l$ from the cylinder draws the string towards him as shown in figure. The cylinder rolls without slipping. The length of the string passed through the hand of the boy while the cylinder reaches his hand is

A system of uniform cylinders and plates is shown in figure. All the cylinders are identical and there is no slipping at any contact. Velocity of lower & upper plate is $V\text{ and }2V$ respectively as shown in figure. Then the ratio of angular speed of the upper cylinders to lower cylinders is

A solid sphere, a hollow sphere and a disc all having the same mass and radius, are placed at the top of an incline and released. The friction coefficients between the objects and the incline are same and not sufficient to allow pure rolling. The least time will be taken in reaching the bottom by:

A sphere $S$ rolls without slipping, moving with a constant speed on a plank $P$. The friction between the upper surface of $P$ and the sphere is sufficient to prevent slipping, while the lower surface of $P$ is smooth and rests on the ground. Initially, $P$ is fixed to the ground by a pin $N$. If $N$ is suddenly removed,