Question

The density of water at the surface of the ocean is $\rho$. If the bulk modulus of water is B, what is the density of ocean water at a depth where the pressure is $n{P}_0$, where $P_0$ is the atmospheric pressure?

• A. $\frac{\rho B}{B-(n-1)P_0}$
• B. $\frac{\rho B}{B+(n-1)P_0}$
• C. $\frac{\rho B}{B-n{P}_0}$
• D. $\frac{\rho B}{B+n{P}_0}$

Answer 1

The correct option is A

$\frac{\rho B}{B-(n-1)P_0}$

Pressure at the surface of the ocean $=P_0$
Pressure at the given depth $=n{P}_0$
$\therefore$ Increase in pressure, $\Delta P=n{P}_0-P_0=(n-1)P_0$
Let V be the volume of a certain mass M of water at the surface, so that $M=\rho V$.
The decrease in volume under pressure $\Delta P$ is $\Delta V=\frac{V\Delta P}{B}$
$\therefore$ Volume of the same mass M of water at the given depth is $V^{\prime }=V-\Delta V=V-\frac{V\Delta P}{B}=V(1-\frac{\Delta P}{B})=\frac{V}{B}(B-\Delta P)$
$\therefore$ Density of water at that depth is ${\rho }^{\prime }=\frac{M}{V^{\prime }}=\frac{\rho V}{V^{\prime }}=\frac{\rho V}{\frac{V}{B}(B-\Delta P)}=\frac{\rho B}{B-\Delta P}=\frac{\rho B}{B-(n-1)P_0}$
Hence, the correct choice is (a).