Find the minimum height $h$ of the obstacle so that the sphere of radius $R$ can stay in equilibrium on an inclined plane of angle $\theta$.

$A$ thin hollow cylinder slides with velocity $v$ without rotating. It then rolls without slipping with the same velocity $v$. Find the ratio of the kinetic energies in the two cases.

The graph between angular momentum $L$ and angular velocity $\omega$ is:

State whether the following statements are true or false:
$(1)$ Torque produces angular velocity in an object.
$(2)$ For the rotational motion of a rigid body,the linear variables of all its particles are the same.

An object has a moment of inertia of $3 \ kg \cdot m^2$. It is rotating with an angular velocity of $2 \ rad/s$. If a mass of $12 \ kg$ is moving with a velocity of $v \ m/s$,at what value of $v$ will their kinetic energies be equal?

$A$ thin string is wrapped around the circumference of a wheel of radius $r$. The wheel has a horizontal axle and a moment of inertia $I$ about it. $A$ weight $mg$ is attached to the end of the string,which falls from rest. After falling through a distance $h$,the angular velocity of the wheel will be: