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:

A body of mass $m$ and radius $r$ is rotated with angular velocity $\omega$ as shown/in the figure $\&$ kept on a surface that has sufficient friction then the body will move :

As shown in the figure, a disc of mass $\mathrm{m}$ is rolling without slipping with an angular velocity $\omega$. The portion $\mathrm{AB}$ is rough and $\mathrm{BC}$ is smooth. When it crosses point $\mathrm{B}$ disc will be in :

A disc is performing pure rolling on a smooth stationary surface with constant angular velocity as shown in figure. At any instant, for the lower most point of the disc