$A$ uniform rod $AB$ of mass $m$ and length $l$ is at rest on a smooth horizontal surface. An impulse $J$ is applied to the end $B$,perpendicular to the rod in the horizontal direction. Find the speed of particle $P$ at a distance $\frac{l}{6}$ from the centre towards $A$ of the rod after time $t = \frac{\pi m l}{12 J}$.

$A$ uniform circular disc of radius $R$, lying on a frictionless horizontal plane, is rotating with an angular velocity $\omega$ about its own axis. Another identical circular disc is gently placed on top of the first disc coaxially. The loss in rotational kinetic energy due to friction between the two discs, as they acquire a common angular velocity, is ($I$ is the moment of inertia of the disc).

One of two identical cylinders,cylinder $A$,rotates at an angular speed of $50 \text{ revolutions per second}$. This rotating cylinder is brought into contact with a second stationary cylinder,$B$. Due to kinetic friction between the two cylinders,the stationary cylinder starts rotating with an angular acceleration,while cylinder $A$ undergoes angular deceleration. If the magnitude of the angular acceleration for both cylinders is $1 \text{ revolution per second}^2$,after how many seconds $(t)$ will the angular speeds of both cylinders become equal?

The total kinetic energy of a body of mass $10 \ kg$ and radius $0.5 \ m$ moving with a velocity of $2 \ m/s$ without slipping is $32.8 \ J$. The radius of gyration of the body is .......... $m$.

$A$ uniform metallic rod rotates about its perpendicular bisector with constant angular speed. If it is heated uniformly to raise its temperature slightly,what happens to its speed of rotation?

$A$ thin uniform rod,pivoted at $O$,is rotating in the horizontal plane with constant angular speed $\omega$,as shown in the figure. At time $t = 0$,a small insect of mass $m$ starts from $O$ and moves with constant speed $v$ with respect to the rod towards the other end. It reaches the end of the rod at time $t = T$ and stops. The angular speed of the system remains $\omega$ throughout. The magnitude of the torque $(|\vec{\tau}|)$ on the system about $O$,as a function of time,is best represented by which plot?