For the situations given in column I match the corresponding values of moment of inertia about an axis passing through $O$.

	Column I (Axis of rotation)		Column II (Moment of inertia about axis of rotation)
(i)	A rod of mass $m$ length $3l$ is bent at $O$.	(a)	$\frac{7}{3}m{l}^2$
(ii)	$\mathrm{L}$-shaped composite rod comprising two identical rods, each of mass $m$ and length $l$.	(b)	$\frac{17}{12}m{l}^2$
(iii)	A rod of mass $m$ and length $l$ is rotating about an axis passing through $O$.	(c)	$m{l}^2$
(iv)	Two identical rods, each of mass $m$ and length $l$, are welded as shown.	(d)	$\frac{55}{48}m{l}^2$

A solid cylinder with $r=0.1 \mathrm{m}$ and mass $M=2 \mathrm{kg}$ is placed such that it is in contact with the vertical and a horizontal surface as shown in figure. The coefficient of static friction is $\mu =(1/3)$ for both the surfaces. Find the distance $d$ (in $\mathrm{cm}$) from the centre of the cylinder at which a force $F=40 \mathrm{N}$ should be applied vertically so that the cylinder just starts rotating in anticlockwise direction.

A uniform rod $AB$ of mass $2 \mathrm{kg}$ is hinged at one end $A.$ The rod is kept in the horizontal position by a massless string tied to point $B.$ Find the reaction of the hinge (in $N$) on end $A$ of the rod at the instant when string is cut. $\left(g=10 \mathrm{m}/{\mathrm{s}}^2\right)$

A square plate $ABCD$ of mass $m$ and side $l$ is suspended with the help of two ideal strings $P$ and $Q$ as shown. Determine the acceleration (in $m/{\mathrm{s}}^2$) of corner $A$ of the square just at the moment the string $Q$ is cut. $\left(g=10 \mathrm{m}/{\mathrm{s}}^2\right)$

A uniform cylinder rests on a cart as shown. The coefficient of static friction between the cylinder and the cart is $0.5.$ If the cylinder is $4 \mathrm{cm}$ in diameter and $10 \mathrm{cm}$ in height, then what is the minimum acceleration (in $m{s}^{-2}$) of the cart needed to cause the cylinder to tip over?

A thin uniform rod $AB$ of mass $1 \mathrm{kg}$ moves translationally with acceleration $a=2 \mathrm{m}/{\mathrm{s}}^2$ due to two antiparallel forces $F_1$ and $F_2.$ The distance between the points at which these forces are applied is equal to $l=0.2 \mathrm{m}.$ If force $F_1=8 \mathrm{N},$ then find the length of the rod in meter.

In figure, a sphere of radius $2 \mathrm{m}$ rolls on a plank. The accelerations of the sphere and the plank are indicated. Find the value of $\alpha$ in $rad/s^2.$

Two rods each of mass $m=4 \mathrm{kg}$ and length $l=1 \mathrm{m}$ are in equilibrium being smoothly hinged at $P.$ Find the tension in the connecting string as shown in figure.

A cylinder is in rotational equilibrium on an inclined plane, inclined at angle $\theta =53°.$ Find the minimum $\mu$ between the wooden log and the inclined plane. (Given: $\sin 53°=\frac{4}{5}$ and $\cos 53°=\frac{3}{5}$)