A monoatomic gas at pressure '$$\mathrm{P}$$' having volume '$$\mathrm{V}$$' expands isothermally to a volume $$2 \mathrm{~V}$$ and then adiabatically to a volume $$16 \mathrm{~V}$$. The final pressure of the gas is $$\left(\gamma=\frac{5}{3}\right)$$

A particle moves in a circular orbit of radius '$$r$$' under a central attractive force, $$F=-\frac{k}{r}$$, where $$\mathrm{k}$$ is a constant. The periodic time of its motion is proportional to

A black reactangular surface of area '$$a$$' emits energy '$$\mathrm{E}$$' per second at $$27^{\circ} \mathrm{C}$$. If length and breadth is reduced to $$\left(\frac{1}{3}\right)^{\text {rd }}$$ of initial value and temperature is raised to $$327^{\circ} \mathrm{C}$$ then energy emitted per second becomes

A closed organ pipe of length '$$\mathrm{L}_c$$' and an open organ pipe of length '$$\mathrm{L}_{\mathrm{o}}$$' contain different gases of densities '$$\rho_1$$' and '$$\rho_2$$' respectively. The compressibility of the gases is the same in both the pipes. The gases are vibrating in their first overtone with the same frequency. What is the length of open organ pipe?