Question

The density of water at the surface of ocean is $\rho$ . If the bulk modulus of water is $B$ , then the density of ocean water at depth, when the pressure is $\alpha {\text{p}}_0$ where $p_0$ is the atmospheric pressure, is

• A. $\frac{\rho \text{B}}{\text{B}-\left(\alpha -1\right)p_0}$
• B. $\frac{\rho B}{B+\left(\alpha -1\right)p_0}$
• C. $\frac{\rho B}{B-\alpha p_0}$
• D. $\frac{\rho B}{B+\alpha p_0}$

Answer 1

Answer: (A)

From the definition of bulk modulus
$\text{B}=-\frac{\text{dp}}{\frac{\text{dV}}{\text{V}}}$
As we move from surface to place where pressure changes to $\alpha {\text{p}}_0$ , let us assume volume changes by $\Delta \text{V}$ , then
$\text{B}=-\frac{\text{V}\Delta \text{p}}{\Delta \text{V}}=\frac{\text{V}\left(\alpha -1\right){\left(\text{p}\right)}_0}{\Delta \text{V}}$
New volume, ${\left(\text{V}\right)}^{{}^{\prime }}=\text{V}-\Delta \text{V}=\left[1-\frac{\left(\alpha -1\right){\left(\text{p}\right)}_0}{\text{B}}\right]V$
Density at the given depth, ${\rho }^{{}^{\prime }}=\rho \frac{\text{V}}{{\text{V}}^{{}^{\prime }}}$ , where $\rho$ is density at surface
${\left(\rho \right)}^{{}^{\prime }}=\frac{\rho \times \text{B}}{\text{B}-\left(\alpha -1\right){\left(\text{p}\right)}_0}$