' $N$ ' molecules of gas $A$, each having mass ' $m$ ' and ' 2 N ' molecules of gas B , each of mass ' 2 m ' are contained in the same vessel which is at constant temperature ' T '. The mean square velocity of $B$ is $V^2$ and mean square of x -component of A is $\omega^2$. The value of $\frac{\omega^2}{\mathrm{~V}^2}$ is

The $\mathrm{p}-\mathrm{V}$ diagram for a fixed mass of an ideal gas undergoing cyclic process is as shown in figure. AB represents isothermal process and CA represents adiabatic process. Which one of the following graphs represents the p-T diagram of this cyclic process?

Two cylinders A and B fitted with piston contain equal amount of an ideal diatomic as at temperature ' T ' K . The piston of cylinder A is free to move while that of B is held fixed. The same amount of heat is given to the gas in each cylinder. If the rise temperature of the gas in A is ' $\mathrm{dT}_{\mathrm{A}}$ ', then the rise in temperature of the gas in cylinder B is $\left(\gamma=\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{C}_{\mathrm{v}}}\right)$

A metal rod having coefficient of linear expansion $2 \times 10^{-5} /^{\circ} \mathrm{C}$ is 0.75 m long at $45^{\circ} \mathrm{C}$. When the temperature rises to $65^{\circ} \mathrm{C}$, the increase in length of the rod will be