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 $..........\,P$.

$A$ swimming pool has a depth of $22 \ m$ and area $700 \ m^2$. Calculate the fractional change $\frac{\Delta V}{V}$ of water at the bottom of the swimming pool. Given that the bulk modulus of water is $2.2 \times 10^9 \ N \ m^{-2}$,$g = 10 \ m \ s^{-2}$,and the density of water is $1000 \ kg \ m^{-3}$.

If the Bulk modulus of lead is $8.0 \times 10^9 \, N/m^2$ and the initial density of the lead is $11.4 \, g/cc$,then under the pressure of $2.0 \times 10^8 \, N/m^2$,the density of the lead is ............. $g/cc$.

If a brass sphere of radius $36 \ cm$ is submerged in a lake at a depth where the pressure is $10^7 \ Pa$,then the change in the radius of the sphere is $($Bulk modulus of brass $= 60 \ GPa)$.

When a sphere is taken to the bottom of a sea of depth $1 \ km$,it contracts in volume by $0.01 \%$. The bulk modulus of the material of the sphere is (Acceleration due to gravity $= 10 \ m \ s^{-2}$,density of sea water $= 10^3 \ kg \ m^{-3}$)

$A$ glass slab is subjected to a pressure of $10 \, atm$. The fractional change in its volume is (Bulk modulus of glass $= 37 \times 10^9 \, N \, m^{-2}$,$1 \, atm = 1 \times 10^5 \, N \, m^{-2}$)