A mass $$'m'$$ is supported by a massless string wound around a uniform hollow cylinder of mass $$m$$ and radius $$R.$$ If the string does not slip on the cylinder, with what acceleration will the mass fall or release?

A bob of mass $$m$$ attached to an inextensible string of length $$l$$ is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed $$\omega \,rad/s$$ about the vertical. About the point of suspension:

A hoop of radius $$r$$ and mass $$m$$ rotating with an angular velocity $${\omega _0}$$ is placed on a rough horizontal surface. The initial velocity of the center of the hoop is zero. What will be the velocity of the center of the hoop when it cases to slip?

A pulley of radius $$2$$ $$m$$ is rotated about its axis by a force $$F = \left( {20t - 5{t^2}} \right)$$ newton (where $$t$$ is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is $$10kg$$-$${m^2}$$ the number of rotation made by the pulley before its direction of motion is reversed, is: