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 $\alpha \textit{p}_{0}$ where $p_{0}$ is the atmospheric pressure, is

• A. $\frac{\rho \textit{B}}{\textit{B} - \left(\alpha - 1\right) p_{0}}$
• 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, $\left(\text{V}\right)^{'} = \text{V} - \Delta \text{V} = \left[1 - \frac{\left(\alpha - 1\right) \left(\text{p}\right)_{0}}{\text{B}}\right]V$
Density at the given depth, $\rho ^{'} = \rho \frac{\text{V}}{\text{V}^{'}}$ , where $\rho $ is density at surface
$\left(\rho \right)^{'} = \frac{\rho \times \text{B}}{\text{B} - \left(\alpha - 1\right) \left(\text{p}\right)_{0}}$