$A$ string is wrapped around the rim of a wheel of moment of inertia $0.40 \ kg \cdot m^2$ and radius $10 \ cm$. The wheel is free to rotate about its axis. Initially,the wheel is at rest. The string is now pulled by a force of $40 \ N$. The angular velocity of the wheel after $10 \ s$ is $x \ rad/s$,where $x$ is $\qquad$

$A$ pulley of radius $1.5\,m$ is rotated about its axis by a force $F = (12t - 3t^2)\,N$ applied tangentially (where $t$ is measured in seconds). If the moment of inertia of the pulley about its axis of rotation is $4.5\,kg\cdot m^2$,the number of rotations made by the pulley before its direction of motion is reversed will be $\frac{K}{\pi}$. The value of $K$ is $.....$

Obtain the relation between angular momentum of a particle and torque acting on it.

What does the time rate of change of angular momentum represent?

$A$ flywheel of moment of inertia $0.4 \, kg \, m^2$ and radius $0.2 \, m$ is free to rotate about a central axis. If a string is wrapped around it and it is pulled with a force of $10 \, N$,then the angular velocity after $4 \, s$ will be ......... $rad \, s^{-1}$.

What constant force must be applied tangentially at the equator to stop the Earth's rotation in one day?