A thin uniform rod of mass $m$ and length $L$ is moving translationally with acceleration $a$ due to two antiparallel forces, as shown. One of the forces is $F$. Find the maximum value of acceleration for which the rod may move translationally.

Figure shows a lamina in the $xy$-plane. Two axes $z$ and $z\text{'}$, pass perpendicular to its plane. Force $F$ acts on the plane of lamina at point $F$, as shown. Which of the following options are true? (Point $P$ is closer to the $z\text{'}$-axis than the $z$-axis.)

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