Consider a fixed wheel of radius $R$ . A small wheel (in the form, of a uniform solid disc) $r$of radius  is performing pure rolling on periphery of bigger wheel. centers of bigger wheel and smaller wheel are joined by rigid rod such that smaller wheel can rotate freely with respect to its centre. Rod joining the (centres is rotating with constant angular velocity ${\omega }_o$ . Whole situation is shown in figure. Choose the correct option(s): (Use $R=4r$)

A uniform rod $AB$ of mass $M$ is attached to a hinge at one end $A$ , and released from rest from the horizontal position. The rod rotates about $A$ , and when it reaches the vertical position the rod strikes a sphere of mass $m$ and radius $r$ initially at rest on the smooth horizontal surface as shown in the adjacent figure. The impact is along the horizontal direction and perfectly elastic. If at the moment of impact, the lowest end of the rod is very close to the smooth horizontal surface. After the impact, the sphere moves along the horizontal and the rod, subsequently rises to a maximum of $60°$ with the vertical Choose the correct statement(s) from the following, taking into account the information given above.
The length of the rod equals $\sqrt{2}r,\left(r=\frac{6\sqrt{2}}{10}\mathrm{m}\right)$
 

Two identical blocks are resting on supports as shown. Two identical bullets travelling with the same speed hit the blocks and get embedded in them in very short time. The blocks leave the contact immediately after bullet strikes them. In case $\left(\mathrm{i}\right)$ bullet's velocity is toward centre of mass and in case $\left(\mathrm{i}\mathrm{i}\right)$ bullet's velocity is off centre. The kinetic energies after collision are $K_1$ & $K_2$ and maximum height reached by the centre of mass are $h_1$ & $h_2$ respectively. Then

A wheel $\mathrm{A}$ is connected to a second wheel $\mathrm{B}$ by means of inextensible string, passing over a pulley $\mathrm{C}$, which rotates about a fixed horizontal axle $\mathrm{O}$, as shown in the figure. The system is released from rest. The wheel $\mathrm{A}$ rolls down the inclined plane $\mathrm{OK}$ thus pulling up wheel $\mathrm{B}$ which rolls along the inclined plane $\mathrm{ON}$ Determine the velocity (in $\mathrm{m}/\mathrm{s}$ ) of the axle of wheel $\mathrm{A}$, when it has travelled a distance $s=3.5 \mathrm{m}$ down the slope. Both wheels and the pulley are assumed to be homogeneous disks of identical weight and radius. Neglect the weight of the string. The string does not slip over $\mathrm{C}$. [Take $\alpha =53°$ and $\beta =37°$ ]

Two forces of magnitude $F$ are acting on a uniform disc kept on a horizontal rough surface as shown in the figure. Friction force by the horizontal surface on the disc is $n F$. Find the value of $n$

We apply a force of $10 N$ on a cord wrapped around a cylinder of mass $2 kg$. The cylinder rolls without slipping on the floor. What is its kinetic energy (in joule) when cylinder has moved by a distance of $3/5 \mathrm{m}$.

A solid cylinder $C$ and a hollow pipe $P$ of same diameter are in contact when they are released from rest as shown in the figure on a long incline plane. Cylinder $C$ and pipe $P$ roll without slipping. Determine the clear gap (in $m$ ) between them after $2\sqrt{3}$seconds

A smooth ring of mass $m$ and radius $R=1 \mathrm{m}$ is pulled at $P$ with a constant acceleration $a=4 \mathrm{m}{ \mathrm{s}}^{-2}$ on a horizontal surface such that the plane of the ring lies on the surface. Find the angular acceleration of the ring at the given position. (in $\mathrm{rad} {\mathrm{s}}^{-2}$ )