Q: The density of water at the surface of an ocean is ρ . If the bulk modulus of water is B, then what is the density of ocean water at a depth where the pressure is *np*_{0}, where *p*_{0} is the atmospheric pressure ?

(a) $ \displaystyle \frac{\rho B}{B + np_0} $

(b) $ \displaystyle \frac{\rho B}{B +( n-1)p_0} $

(c) $ \displaystyle \frac{\rho B}{B – n p_0} $

(d) $ \displaystyle \frac{\rho B}{B – ( n-1)p_0} $

Ans: (d)

Sol: Increase in pressure (∆p)= n p_{0} – p_{0} = (n-1)p_{0}

Let V be the volume of a certain mass m of water at the surface, so that m= ρV.

The decrease in volume under the pressure ∆p is

∆V = V∆p/B

Volume of the same mass m of water at the given depth is

V’ = V – ∆V = V-V∆p/B

= V(1- ∆p/B) = V/B (B- ∆p)

Density of water at that depth is

ρ’= m/V’ = ρV/V’= ρV/(V/B (B- ∆p))

ρ’= ρB/(B- ∆p)= ρB/(B-(n-1)p_{0} )