Under isothermal condition,the pressure of a gas is given by $P = aV^{-3}$,where $a$ is a constant and $V$ is the volume of the gas. The bulk modulus at constant temperature is equal to $..........\,P$.

$A$ certain pressure $P$ is applied to $1 \text{ litre}$ of water and $2 \text{ litre}$ of a liquid separately. Water gets compressed to $0.01 \%$ whereas the liquid gets compressed to $0.03 \%$. The ratio of Bulk modulus of water to that of the liquid is $\frac{3}{x}$. The value of $x$ is $...........$

The compressibility of water per unit atmospheric pressure is $\sigma$. If there is a decrease in volume $V$ due to an applied pressure $P$,find the change in volume.

$A$ spherical ball contracts in volume by $0.02 \%$,when subjected to a normal uniform pressure of $50$ atmosphere. The Bulk modulus of its material is ............. $Nm^{-2}$.

$A$ copper solid cube of $60\, mm$ side is subjected to a pressure of $2.5 \times 10^7\, Pa$. If the bulk modulus of copper is $1.25 \times 10^{11}\, N/m^2$,the change in the volume of the cube is:

If the average depth of an ocean is $4000 \ m$ and the bulk modulus of water is $2 \times 10^9 \ N m^{-2}$,then the fractional compression $\frac{\Delta V}{V}$ of water at the bottom of the ocean is $\alpha \times 10^{-2}$. The value of $\alpha$ is . . . . . . (Given,$g=10 \ m s^{-2}, \rho=1000 \ kg m^{-3}$)