Question

the density $p$ of water of bulk modulus $B$ at a depth $y$ in the ocean is related to the density at surface ${\rho }_0$ by the relation.

• A. $\rho ={\rho }_0\left[1-\frac{{\rho }_{0}gv}{B}\right]$
• B. $\rho ={\rho }_0\left[1+\frac{{\rho }_{0}gv}{B}\right]$
• C. $\rho ={\rho }_0\left[1-\frac{B}{{\rho }_{0}hgy}\right]$
• D. $\rho ={\rho }_0\left[1-\frac{B}{{\rho }_{0}gy}\right]$

Answer 1

The correct option is B $\rho ={\rho }_0\left[1+\frac{{\rho }_{0}gv}{B}\right]$

$\begin{array}{c}\text{Denspty of water at surface}={\rho }_0\\ \text{Denspty of water at some depth}=\rho \\ \text{Bulk modulus=}B\\ \text{Depth}=y\\ \text{inetial volume}=v_0\end{array}$

$\begin{array}{c}\text{inetial volume}=v_0\\ \text{Bulk modulus}B=\frac{-dP}{\frac{dv}{V_0}}\\ \text{Let}v\text{be the final volume}\\ \text{dp represents the change in pressure}\end{array}$

$\begin{array}{c}\text{we know that change in pressure between Surface and a}\\ \text{a depth}y\text{is}y{\rho }_{0}g\end{array}$

$\begin{array}{cc}\Rightarrow & dp=y{\rho }_{0}g\\ \Delta v& =-v_0\frac{dP}{B}\\ v-v_0& =-v_0\frac{dP}{B}\\ v& =v_0[1-\frac{dP}{B}]\end{array}$

$\begin{array}{cc}\therefore \text{Density}& =\rho ={\rho }_0{[1-\frac{dp}{B}]}^{-1}\\ \text{For smaller values of}\frac{dp}{B}& \\ & \rho ={\rho }_0[1+\frac{dP}{B}]\\ & f={\rho }_0[1+\frac{{\rho }_{0}gy}{B}]\end{array}$