The centre of a wheel rolling on a plane surface moves with a speed $v_0$. A particle on the rim of the wheel at the same level as the centre will be moving at the speed

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

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 large spool of rope lies on the ground as shown in the fig: The end, labelled $X$ is pulled a distance $S$ in the horizontal direction. The spool rolls without slipping The centre of mass of the spool moves a. distance

A thin string is wrapped several times around a cylinder kept on a rough horizontal surface. A boy standing at a distance $\ell$ 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 sphere $\mathrm{S}$ rolls without slipping, moving with a constant speed on a plank $\mathrm{P}$. The friction between the upper surface of $\mathrm{P}$ and the sphere is sufficient to prevent slipping; while the lower surface of $\mathrm{P}$ is smooth and rests on the ground. Initially, $\mathrm{P}$ is fixed to the ground by a pin $\mathrm{N}$. If $\mathrm{N}$ is suddenly removed: