$A$ non-uniform cylinder of mass $m$,length $l$,and radius $r$ has its center of mass at a distance $l/4$ from the geometric center,lying on the axis of the cylinder. The cylinder is kept in a liquid of uniform density $\rho$. The moment of inertia of the cylinder about its center of mass is $I$. The angular acceleration of the cylinder just after it is released from the horizontal position shown in the figure is:

$A$ hinged construction consists of three rhombs with the side ratio $5:3:2$. Vertex $A_3$ moves in the horizontal direction at a velocity $v$. The velocity of $A_2$ is ....... $V$.

$A$ plank with a uniform sphere placed on it rests on a smooth horizontal plane. The plank is pulled to the right by a constant force $F$. If the sphere does not slip over the plank,which of the following statements is incorrect?

$A$ thin hollow cylinder slides with velocity $v$ without rotating. It then rolls without slipping with the same velocity $v$. Find the ratio of the kinetic energies in the two cases.

$A$ block of mass $M$ has a circular cut with a frictionless surface as shown. The block rests on the horizontal frictionless surface of a fixed table. Initially,the right edge of the block is at $x=0$,in a coordinate system fixed to the table. $A$ point mass $m$ is released from rest at the topmost point of the path as shown and it slides down. When the mass loses contact with the block,its position is $x$ and the velocity is $v$. At that instant,which of the following options is/are correct?
$[A]$ The $x$ component of displacement of the center of mass of the block $M$ is: $-\frac{m R}{M+m}$.
$[B]$ The position of the point mass is: $x=-\sqrt{2} \frac{mR}{M+m}$.
$[C]$ The velocity of the point mass $m$ is: $v=\sqrt{\frac{2 g R}{1+\frac{m}{M}}}$.
$[D]$ The velocity of the block $M$ is: $V=-\frac{m}{M} \sqrt{2 g R}$.

$A$ uniform disc is rolling on a horizontal surface. At a certain instant,$B$ is the point of contact and $A$ is the topmost point of the disc,where $R$ is the radius of the disc.