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.

A uniform rod of length $L$ (in between the supports) and mass $m$ is placed on two supports $A$ and $B$. The rod breaks suddenly at length $\frac{L}{10}$ from the support $B$. Find the reaction at support $A$ immediately after the rod breaks.

A rigid body of radius $R$, either hollow or solid, lies on a smooth horizontal surface. The body is pulled by a horizontal force acting tangentially from the highest point. The distance travelled by the body in the time in which it makes one full rotation is the same, that it will make in one full rotation during pure rolling. The rigid body will be