$A$ ring of mass $m$ can freely slide along the massless curved rod as shown. At the lower most point,the curved path becomes vertical. If the whole system is released from rest,what is the velocity of the ring $(v)$ at the lowermost point just before touching the block $M$ (all surfaces are smooth)?

$A$ $3.0\ kg$ bobbin consists of a central cylinder of radius $5.0\ cm$ and two end plates each of radius $6.0\ cm$. It is placed on a slotted incline,where friction is sufficient to prevent sliding. $A$ block of mass $4.5\ kg$ is suspended from a cord wound around the bobbin and passing through the slot under the incline. If the bobbin is in static equilibrium,what is the angle of tilt $\theta$ of the incline?

$A$ homogeneous solid cylindrical roller of radius $R$ and mass $M$ is pulled on a cricket pitch by a horizontal force $F$ applied at its center. Assuming rolling without slipping,the angular acceleration of the cylinder is

$A$ rod of length $1 \ m$ is held vertically. What will be the speed of the other end when it touches the ground without slipping?

$A$ disc rotating about its axis from rest,acquires the angular speed $100 \, rev/s$ in $4 \, s$. The angle rotated by it during these four seconds (in radian) is ...... $\pi$.

Two small balls are fixed at the ends of a light rigid rod of length $L = 0.4 \ m$. The system is released from a height $h = 5 \ m$ with the rod horizontal. The balls collide with the horizontal surface and rebound. The coefficient of restitution between ball $A$ and the ground is $e_A = 0.6$ and that between ball $B$ and the ground is $e_B = 0.4$. Find the angular speed (in $rad/s$) of the rod just after the collision. Take the acceleration due to gravity $g = 10 \ m/s^2$.