The cross-sectional area of a piston is $8 \times 10^{-3} \, m^2$. The gas has a temperature of $300 \, K$,a pressure of $1.0 \times 10^5 \, Pa$,and a volume of $2.4 \times 10^{-3} \, m^3$. When the gas is heated by a heater,the piston moves by $0.1 \, m$. If the spring constant is $8000 \, N/m$,find the final temperature in $K$.

At which temperature will the speed of sound in hydrogen be the same as the speed of sound in oxygen at $100\,^{\circ}C$?

$A$ box of $1 \ m^{3}$ is filled with nitrogen at $1.5 \ atm$ at $300 \ K$. The box has a hole of an area $0.010 \ mm^{2}$. How much time is required for the pressure to reduce by $0.10 \ atm$,if the pressure outside is $1 \ atm$?

The average translational kinetic energy and the $r.m.s.$ speed of molecules in a sample of oxygen gas at $300 \, K$ are $6.21 \times 10^{-21} \, J$ and $484 \, m/s$ respectively. The corresponding values at $600 \, K$ are nearly (assuming ideal gas behaviour):

$A$ fixed thermally conducting cylinder has a radius $R$ and height $L_0$. The cylinder is open at its bottom and has a small hole at its top. $A$ piston of mass $M$ is held at a distance $L$ from the top surface,as shown in the figure. The atmospheric pressure is $P_0$.
$1.$ The piston is now pulled out slowly and held at a distance $2L$ from the top. The pressure in the cylinder between its top and the piston will then be
$(A) P_0$ $(B) \frac{P_0}{2}$ $(C) \frac{P_0}{2} + \frac{Mg}{\pi R^2}$ $(D) \frac{P_0}{2} - \frac{Mg}{\pi R^2}$
$2.$ While the piston is at a distance $2L$ from the top,the hole at the top is sealed. The piston is then released,to a position where it can stay in equilibrium. In this condition,the distance of the piston from the top is
$(A) \left(\frac{2P_0 \pi R^2}{\pi R^2 P_0 + Mg}\right)(2L)$ $(B) \left(\frac{P_0 \pi R^2 - Mg}{\pi R^2 P_0}\right)(2L)$ $(C) \left(\frac{P_0 \pi R^2 + Mg}{\pi R^2 P_0}\right)(2L)$ $(D) \left(\frac{P_0 \pi R^2}{\pi R^2 P_0 - Mg}\right)(2L)$
$3.$ The piston is taken completely out of the cylinder. The hole at the top is sealed. $A$ water tank is brought below the cylinder and put in a position so that the water surface in the tank is at the same level as the top of the cylinder as shown in the figure. The density of the water is $\rho$. In equilibrium,the height $H$ of the water column in the cylinder satisfies
$(A) \rho g(L_0 - H)^2 + P_0(L_0 - H) + L_0 P_0 = 0$
$(B) \rho g(L_0 - H)^2 - P_0(L_0 - H) - L_0 P_0 = 0$
$(C) \rho g(L_0 - H)^2 + P_0(L_0 - H) - L_0 P_0 = 0$
$(D) \rho g(L_0 - H)^2 - P_0(L_0 - H) + L_0 P_0 = 0$
Give the answer for questions $1, 2$ and $3$.

$A$ diatomic gas of molecular mass $40 \, g/mol$ is filled in a rigid container at temperature $30^{\circ} C$. It is moving with velocity $200 \, m/s$. If it is suddenly stopped,the rise in the temperature of the gas is .........