A square plate of mass $M$ and side length $l$ may either rotate in $x‐z$ plane about a point $A$ or may rotate by falling towards $x‐y$ plane. If $I_1$and $I_2$ are moments of inertia in the two cases, respectively, find the correct option.

A door is hinged at one end and is free to rotate about a vertical axis (see the figure). Does its weight cause any torque about this axis? Give a reason for your answer.

Two cylindrical hollow drums of radii $R$ and $2R$ and of common height $h$ are rotating with angular velocities $\text{ω}$ (anticlockwise) and $2\text{ω}$ (clockwise), respectively. Their axes, fixed are parallel and in a horizontal plane separated by $3R+\text{δ}$. They are now brought in contact $\left(\text{δ}\to 0\right)$. 

(a) Show the frictional forces just after contact.
(b) Identify forces and torques external to the system just after contact.
(c) What would be the ratio of final angular velocities when friction ceases?

A cubical block of side $L$ and mass $m$ is placed on a horizontal floor. In the arrangement, as shown, force $F$ is applied at the end of the plane sheet $PQ$, which is firmly attached with the block. What should be the minimum value of $PQ$ so that the block may be tipped about an edge?

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.)