Two samples A and B of gas initially at the same pressure and temperature. They are compressed from volume $V$ to $\dfrac{V}{2}$ (A isothermally and B adiabatically). The relation between pressure of the gas A ($P_A$) and pressure of gas B ($P_B$) is

A vessel has $6$ gram of hydrogen at pressure $P$ and temperature $500$ K. A small hole is made in it so that hydrogen leaks out. How much hydrogen leaks out if the final pressure is $\dfrac{P}{2}$ and the temperature falls to $300$ K?

A particle executes simple harmonic motion between $x = -A$ and $x = +A$. If time taken by particle to go from $x = 0$ to $\dfrac{A}{2}$ is $2$s, then time taken by particle in going from $x = \dfrac{A}{2}$ to A is

There is a body having mass $m$ and performing SHM with amplitude '$a$'. There is a restoring force $F=-Kx$, where $x$ is the displacement. The total energy of the body depends upon which of the following?