The torque of a force $5\hat{i}+3\hat{j}-7\hat{k}$ about the origin is $\tau$. If the force acts on a particle whose position vector is $2\hat{i}+2\hat{j}+\hat{k}$, then the value of $\tau$ will be

Solid sphere $A$ is rotating about an axis $PQ$. If the radius of the sphere is $5 \mathrm{cm}$, then its radius of gyration about $PQ$ will be $\sqrt{x} \mathrm{cm}$. The value of $x$ is _____.

A uniform solid cylinder with radius $R$ and length $L$ has moment of inertia $I_1$, about the axis of cylinder. A concentric solid cylinder of radius $R^{\text{'}}=\frac{R}{2}$  and length $L^{\text{'}}=\frac{L}{2}$ is carved out of the original cylinder. If $I_2$ is the moment of inertia of the carved out portion of the cylinder then $\frac{I_1}{I_2}=$  __________.

(Both $I_1$ and $I_2$ are about the axis of the cylinder)   

An object of mass $8 \mathrm{kg}$ is hanging from one end of a uniform rod $CD$ of mass $2 \mathrm{kg}$ and length $1 \mathrm{m}$ pivoted at its end $C$ on a vertical wall as shown in figure. It is supported by a cable $AB$ such that the system is in equilibrium. The tension in the cable is:

(Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

${\mathrm{I}}_{\mathrm{CM}}$ is moment of inertia of a circular disc about an axis (CM) passing through its center and perpendicular to the plane of disc. ${\mathrm{I}}_{\mathrm{AB}}$ is its moment of inertia about an axis AB perpendicular to plane and parallel to axis CM at a distance $\frac{2}{3}\mathrm{R}$ from center, where $\mathrm{R}$ is the radius of the disc. The ratio of ${\mathrm{I}}_{\mathrm{AB}}$ and ${\mathrm{I}}_{\mathrm{CM}}$ is $x:9$. The value of $x$ is ______.

A particle of mass $100 \mathrm{g}$ is projected at time $t=0$ with a speed $20 \mathrm{m} {\mathrm{s}}^{–1}$ at an angle $45°$ to the horizontal as given in the figure. The magnitude of the angular momentum of the particle about the starting point at time $t=2 \mathrm{s}$ is found to be $\sqrt{K} \mathrm{kg} {\mathrm{m}}^2 {\mathrm{s}}^{-1}$. The value of $K$ is ______.

(Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)