Question

At a particular point in the Pacific ocean, the density of sea water increase non – linearly with depth according to $\rho ={\rho }_0+m{z}^2$, where ${\rho }_0$ is the density at the surface, $z$ is the depth below the surface and $m$ is a constant. The algebraic equation for the relationship between pressure and depth is

• A. $P=P_{atm}+{\rho }_{0}gz+\frac{1}{3}mg{z}^3$
• B. $P=P_{atm}+2{\rho }_{0}gz+\frac{1}{3}mg{z}^3$
• C. $P=P_{atm}+{\rho }_{0}gz+\frac{2}{3}mg{z}^3$
• D. $P=P_{atm}+{\rho }_{0}gz+mg{z}^3$

Answer 1

The correct option is A $P=P_{atm}+{\rho }_{0}gz+\frac{1}{3}mg{z}^3$

Pressure at the surface of pacific ocean $=P_{atm}$
At $z$ depth below the sea level
We know,
$\frac{dP}{dz}=\rho g$
${\int }_{P_{atm}}^{P_z}dp={\int }_0^z\rho gdz$
${\int }_{P_{atm}}^{P_z}dp={\int }_0^{z}g({\rho }_0+m{z}^2)dz$
$P_z-P_{atm}=g({\rho }_{0}z+\frac{m{z}^3}{3})$
$P_z=P_{atm}+g{\rho }_{0}z+g\frac{m{z}^3}{3}$