In the figure shown a ring $A$ is rolling without sliding with a velocity $v$ on the horizontal surface of the body $B$ (of the same mass as $A$). All surfaces are smooth. $B$ has no initial velocity. What will be the maximum height (from initial position) reached by $A$ on $B$.

A uniform rod of mass $m$, length $l$ is placed over a smooth horizontal surface along $y$-axis and is at rest as shown in the figure. An impulsive force $F$ is applied for a small-time $∆t$ along $x$-direction at the point $A$ after this rod moves freely. The $x$–coordinate of the end $A$ of the rod when the rod becomes parallel to $x$–axis for the first time is (initially the coordinate of centre of mass of the rod is $\left(0, 0\right)$

A uniform rectangular plate of mass $m$ which is free to rotate about the smooth vertical hinge passing through the centre and perpendicular to the plate is lying on a smooth horizontal surface. A particle of mass $m$ moving with speed $u$ collides with the plate and sticks to it as shown in the figure. The angular velocity of the plate after the collision will be

A particle of mass $m$ is moving horizontally at speed $v$ perpendicular to a uniform rod of length $d$ and mass $M=6m$. The rod is hinged at centre $O$ and can freely rotate in the horizontal plane about a fixed vertical axis passing through its centre $O$. The hinge is frictionless. The particle strikes and sticks to the end of the rod. The angular speed of the system just after the collision

A uniform sphere of mass $m$ is given some angular velocity about a horizontal axis through its centre and gently placed on a plank of mass $m$. The coefficient of friction between the two is $\mu$. The plank rests on a smooth horizontal surface. The initial acceleration of the centre of the sphere relative to the plank will be

A small block of mass $m$ is rigidly attached at $P$ to a ring of mass $3m$ and radius $r$. The system is released from rest at $\mathrm{\theta }=90°$ and rolls without sliding. The angular acceleration of the ring just after release is