Consider a badminton racket with length scales as shown in the figure. If the mass of the linear and circular portions of the badminton racket are same $(M)$ and the mass of the threads are negligible,the moment of inertia of the racket about an axis perpendicular to the handle and in the plane of the ring at $\frac{r}{2}$ distance from the end $A$ of the handle will be ....... $Mr^2$?

$A$ solid cylinder of mass $2 \ kg$ and radius $0.2 \ m$ is rotating with an angular velocity of $3 \ rad/s$. $A$ particle of mass $0.5 \ kg$ moving with a velocity of $5 \ m/s$ strikes its periphery and sticks to it. The loss in kinetic energy due to the collision is ....... $J$.

$A$ particle starts from the point $(0\,m, 8\,m)$ and moves with a uniform velocity of $3\, \hat{i} \,m/s$. After $5\,s$,the angular velocity of the particle about the origin will be:

$A$ disc of mass $M$ and radius $R$ is free to rotate about its vertical axis as shown in the figure. $A$ battery-operated motor of negligible mass is fixed to this disc at a point on its circumference. Another disc of the same mass $M$ and radius $R/2$ is fixed to the motor's thin shaft. Initially,both the discs are at rest. The motor is switched on so that the smaller disc rotates at a uniform angular speed $\omega$. If the angular speed at which the large disc rotates is $\omega/n$,then the value of $n$ is. . . . .

$A$ sphere of mass $M$ and radius $R$ is attached by a light rod of length $l$ to a point $P$. The sphere rolls without slipping on a circular track as shown. It is released from the horizontal position. The angular momentum of the system about $P$ when the rod becomes vertical is:

For the pivoted slender rod of length $l$ as shown in the figure,the angular velocity as the bar reaches the vertical position after being released from the horizontal position is: