$A$ solid sphere and a disc of the same radius are released from the same height on an inclined plane and they reach the bottom of the plane at different times. This is due to their:

The figure shows a system consisting of $(i)$ a ring of outer radius $3R$ rolling clockwise without slipping on a horizontal surface with angular speed $\omega$ and $(ii)$ an inner disc of radius $2R$ rotating anti-clockwise with angular speed $\omega/2$. The ring and disc are separated by frictionless ball bearings. The system is in the $x-z$ plane. The point $P$ on the inner disc is at distance $R$ from the origin,where $OP$ makes an angle of $30^{\circ}$ with the horizontal. Then with respect to the horizontal surface,
$(A)$ the point $O$ has linear velocity $3R\omega\hat{i}$.
$(B)$ the point $P$ has a linear velocity $\frac{11}{4}R\omega\hat{i} + \frac{\sqrt{3}}{4}R\omega\hat{k}$.
$(C)$ the point $P$ has linear velocity $\frac{13}{4}R\omega\hat{i} - \frac{\sqrt{3}}{4}R\omega\hat{k}$.
$(D)$ The point $P$ has a linear velocity $(3 - \frac{\sqrt{3}}{4})R\omega\hat{i} + \frac{1}{4}R\omega\hat{k}$.

$A$ uniform rod is fixed to a rotating turntable so that its lower end is on the axis of the turntable and it makes an angle of $20^o$ to the vertical. (The rod is thus rotating with uniform angular velocity about a vertical axis passing through one end.) If the turntable is rotating clockwise as seen from above,is there a torque acting on it,and if so,in what direction?

One ice skater of mass $m$ moves with speed $2v$ to the right,while another of the same mass $m$ moves with speed $v$ toward the left,as shown in figure $I$. Their paths are separated by a distance $b$. At $t = 0$,when they are both at $x = 0$,they grasp a pole of length $b$ and negligible mass. For $t > 0$,consider the system as a rigid body of two masses $m$ separated by distance $b$,as shown in figure $II$. Which of the following is the correct formula for the motion after $t = 0$ of the skater initially at $y = b/2$?

During a test of a jet engine,its compressor rotates according to the graph shown. The total number of revolutions completed by the compressor during the test is:

$A$ wheel of radius $r$ rolls without slipping with a speed $v$ on a horizontal road. When it is at a point $A$ on the road,a small lump of mud separates from the wheel at its highest point $B$ and drops at point $C$ on the road. The distance $AC$ will be