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 nP${}_o$ , where P${}_o$ is the atmosphreic pressure:

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

Answer 1

The correct option is A $\frac{\rho B}{B-(n-1)P_o}$

Let the volume of certain mass (M) of water at surface be $V$ $⟹\rho =\frac{M}{V}$

Pressure at a depth $P^{\prime }=n{P}_o$

Thus change in pressure $\Delta P=n{P}_o-P_o=(n-1)P_o$

Bulk modulus $B=\frac{\Delta P}{\Delta V/V}$ $⟹\Delta V=\frac{(n-1)P_{o}V}{B}$

As at a depth, the pressure increases. Thus the volume of water will decrease at that depth.

$\therefore$ Volume of water of mass (M) at that depth $V^{\prime }=V-\Delta V$

$\therefore$ $V^{\prime }=V-\frac{(n-1)P_{o}V}{B}=(B-(n-1\left)P_o\right)\frac{V}{B}$ ..............(1)

Density at that depth ${\rho }^{\prime }=\frac{M}{V^{\prime }}=\rho \frac{V}{V^{\prime }}$

$\therefore$ ${\rho }^{\prime }=\frac{\rho B}{B-(n-1)P_o}$ (using 1)