When a body slides down from rest along a smooth inclined plane making an angle of $30°$ with the horizontal, it takes time $T.$ When the same body slides down from the rest along a rough inclined plane making the same angle and through the same distance, it takes time $\alpha T,$ where $\alpha$ is a constant greater than $1.$ The co-efficient of friction between the body and the rough plane is $\frac{1}{\sqrt{x}}\left(\frac{{\alpha }^2-1}{{\alpha }^2}\right)$ where $x=\_\_\_\_\_\_\_\_\_\_.$

A steel block of $10 \mathrm{kg}$ rests on a horizontal floor as shown. When three iron cylinders are placed on it as shown, the block and cylinders go down with an acceleration $0.2 \mathrm{m} {\mathrm{s}}^{-2}.$ The normal reaction $R^{\text{'}}$ by the floor if mass of the iron cylinders are equal and of $20 \mathrm{kg}$ each is $\left(\text{in} \mathrm{N}\right)$,

[Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$ and $\left.{\mu }_{\mathrm{s}}=0.2\right]$

The length of a metal wire is ${\ell }_1,$ when the tension in it is $T_1$ and is ${\ell }_2$ when the tension is $T_2.$ The natural length of the wire is:

Two billiard balls of equal mass $30 \mathrm{g}$ strike a rigid wall with same speed of $108 \mathrm{km} {\mathrm{h}}^{-1}$ (as shown) but at different angles. If the balls get reflected with the same speed, then the ratio of the magnitude of impulses imparted to ball $a$ and ball $b$ by the wall along $X$ direction is:

A block of mass $m$ slides along a floor while a force of magnitude $F$ is applied to it at an angle $\theta$ as shown in figure. The coefficient of kinetic friction is ${\mu }_K$. Then, the block's acceleration $a$ is given by : ($g$ is acceleration due to gravity)