$A$ body rotates about a fixed axis with an angular acceleration of $3 \, rad/s^2$. The angle rotated by it during the time when its angular velocity increases from $10 \, rad/s$ to $20 \, rad/s$ (in radian) is ..........

$A$ wheel rotates at $720 \ rpm$ about an axis passing through its center. It is brought to rest by applying a constant torque for $8 \ s$. What is the value of the torque in $Nm$? (Given: $I = \frac{24}{\pi} \ kg \cdot m^2$)

$A$ thin uniform rod is pivoted at $O$. It rotates in a horizontal plane with a constant angular velocity $\omega$. At $t = 0$,a small insect starts moving from $O$ towards the other end with a constant speed $v$ relative to the rod. It reaches the other end at $t = T$ and stops. The angular velocity $\omega$ of the system remains constant. Which of the following graphs best represents the magnitude of the torque $|\tau|$ at $O$ as a function of time $t$?

An automobile moves on a road with a speed of $54 \,km h^{-1}.$ The radius of its wheels is $0.45\, m$ and the moment of inertia of the wheel about its axis of rotation is $3\, kg m^2$. If the vehicle is brought to rest in $15\, s,$ the magnitude of average torque transmitted by its brakes to the wheel is .......... $kg \,m^2\, s^{-2}$.

$A$ rope is wound around a hollow cylinder of mass $3\, kg$ and radius $40\, cm.$ What is the angular acceleration (in $rad/s^2$) of the cylinder if the rope is pulled with a force of $30\, N$?

$A$ wheel is rotating at a rate of $900$ revolutions per minute about its axis. When the power source is cut off,it comes to rest in $1$ minute. The angular retardation of the wheel in $rad/s^2$ is: