A cylinder and a variable mass $M$ are arranged on a fixed wedge, using a light string and a massless pulley. There is enough friction between the cylinder and the wedge to prevent any slipping.

Two identical discs of mass $m$ and radius $r$ are arranged, as shown in the figure. If $\alpha$ is the angular acceleration of the lower disc and $a_{cm}$ is the linear acceleration of the centre of mass of the lower disc, then the relation between $a_{cm}$, $\alpha$ and $r$ is

A uniform thin rod of mass $m$ and length $l$ is held horizontally by two vertical strings, attached to the two ends. One of the string is cut. Find the angular acceleration soon after it is cut.

A uniform disc of mass $M$ and radius $R$ is released from the shown position. $PQ$ is a string, $OP$ is a horizontal line, $O$ is the centre of the disc and distance$OP$ is $\frac{R}{2}$. Then, the tension in the string just after the disc is released will be

A solid cylinder of radius $r$ and mass $m$ is placed with no initial velocity on a moving belt $B$. It is placed against a rough wall $A$, which restricts the sliding of the cylinder on the belt, as shown in the figure. Assuming that the coefficient of friction at $A$ and $B$ is $\mathrm{\mu }(<1)$, the angular acceleration $\mathrm{\alpha }$ of the cylinder will be

A pulley of radius $r$ can rotate about a horizontal axis and its moment of inertia about its axis is $I$. A light string fixed to the edge of the pulley is wound around it several times, then hangs vertically, passes under a light pulley of radius $\frac{r}{2}$ and then rises vertically. Its other end is attached to the centre of the larger pulley. A particle of mass $m$ is attached to the light pulley. The acceleration of the particle is

A uniform rod $\mathrm{AB}$ of mass $m$ is suspended in equilibrium from two similar springs, as shown. If spring $2$ breaks, then the angular acceleration of the bar at that instant will be

A uniform pole of length $L$ and mass $M$ is pivoted on the ground with a frictionless hinge $O$. The pole is free to rotate without friction, about a horizontal axis passing through $O$ and normal to the plane of the page. The pole makes an angle $\mathrm{\theta }$ with the horizontal. The pole is released from rest in the position shown, then the linear acceleration of the free end $P$ of the pole, just after its release would be

One end of a thin bar $AB$ of mass $m$ and length $l$ is hanging from a massless string. The other end is held at an angle of $45°$ with horizontal. Find the tension in the string, immediately after the release of the bar.