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 np0, where p0 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 p0 – p0 = (n-1)p0
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)p0 )