$A$ circular disc of mass $M$ and radius $R$ is rotating about its axis with angular speed $\omega_{1}$. If another stationary disc having radius $\frac{R}{2}$ and same mass $M$ is dropped co-axially onto the rotating disc,gradually both discs attain a constant angular speed $\omega_{2}$. The energy lost in the process is $p \%$ of the initial energy. The value of $p$ is:

$A$ thin uniform circular disc of mass $M$ and radius $R$ is rotating with angular velocity $\omega$ in a horizontal plane about an axis passing through its centre and perpendicular to its plane. Another disc of the same radius but of mass $\frac{M}{2}$ is placed gently on the first disc co-axially. The new angular velocity will be

$A$ person standing on a rotating platform has his hands lowered. He suddenly outstretch his arms. The angular momentum

$A$ bullet of mass $10 \; g$ and speed $500 \; m/s$ is fired into a door and gets embedded exactly at the centre of the door. The door is $1.0 \; m$ wide and weighs $12 \; kg$. It is hinged at one end and rotates about a vertical axis practically without friction. Find the angular speed of the door just after the bullet embeds into it.

$A$ horizontal circular platform of radius $R = 0.5 \ m$ and mass $M = 0.45 \ kg$ is free to rotate about its axis. Two massless spring toy-guns,each carrying a steel ball of mass $m = 0.05 \ kg$,are attached to the platform at a distance $r = 0.25 \ m$ from the centre on its either sides along its diameter (see figure). Each gun simultaneously fires the balls horizontally and perpendicular to the diameter in opposite directions. After leaving the platform,the balls have a horizontal speed of $v = 9 \ m/s$ with respect to the ground. The rotational speed of the platform in $rad/s$ after the balls leave the platform is:

$A$ uniform disc of mass $100 \ kg$ and radius $2 \ m$ is rotating at $1 \ rad/s$ about a perpendicular axis passing through its centre. $A$ boy of mass $60 \ kg$ standing at the centre of the disc suddenly jumps to a point which is $1 \ m$ from the centre of the disc. The final angular velocity of the boy (in $rad/s$) is