$A$ homogeneous solid cylindrical roller of radius $R$ and mass $M$ is pulled on a cricket pitch by a horizontal force $F$ applied at its center. Assuming rolling without slipping,the angular acceleration of the cylinder is

The pulley shown in the figure is made using a thin rim and two rods of length equal to the diameter of the rim. The rim and each rod have a mass of $M$. Two blocks of mass $M$ and $m$ are attached to two ends of a light string passing over the pulley, which is hinged to rotate freely in a vertical plane about its centre. The magnitude of the acceleration experienced by the blocks is . . . . . . (assume no slipping of the string on the pulley.)

$A$ uniform rod of length $l$ is being rotated in a horizontal plane with a constant angular speed $\omega$ about an axis passing through one of its ends. If the tension generated in the rod due to rotation is $T(x)$ at a distance $x$ from the axis,then which of the following graphs depicts it most closely?

$A$ ring of mass $m$ can freely slide along the massless curved rod as shown. At the lower most point,the curved path becomes vertical. If the whole system is released from rest,what is the velocity of the ring $(v)$ at the lowermost point just before touching the block $M$ (all surfaces are smooth)?

$A$ mass $m$ hangs with the help of a string wrapped around a pulley on a frictionless bearing. The pulley has mass $m$ and radius $R$. Assuming the pulley to be a perfect uniform circular disc,the acceleration of the mass $m$,if the string does not slip on the pulley,is

$A$ rigid body rotates about a fixed axis with variable angular velocity equal to $\alpha - \beta t$,where $t$ is time and $\alpha, \beta$ are constants. Find the angle (in radians) through which it rotates before it stops.