$A$ 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 having a moment of inertia of $4 \,kg \cdot m^2$ about its symmetrical axis rotates at a rate of $240 \,rpm$ about it. The torque required to stop the rotation of the wheel in one minute is ............ $N \cdot m$.

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 .......... $kg \,m^2\, s^{-2}$.

$A$ disc of mass $M \ kg$ and radius $R \ m$ is rotating at an angular speed of $\omega \ rad/s$ when the motor is switched off. Neglecting the friction at the axle,the force that must be applied tangentially to the wheel to bring it to rest in time $t$ is ...............

$A$ rod of mass $M$ and length $L$ is placed in a horizontal plane with one end hinged about a vertical axis. $A$ horizontal force of $F = \frac{Mg}{2}$ is applied at a distance $\frac{5L}{6}$ from the hinged end. The angular acceleration of the rod will be:

$A$ torque of $50 \ N \cdot m$ causes a wheel to rotate from rest through $200 \ rad$ in $5 \ s$. The value of the angular acceleration produced is ...... $rad/s^2$.