Question

If the average depth of an ocean is 4000 m and the bulk modulus of water is $2 \times 10^9 \, \text{N/m}^2$, then the fractional compression $\frac{\Delta V}{V}$ of water at the bottom of the ocean is $\alpha \times 10^{-2}$. The value of $\alpha$ is ______ (Given $g = 10 \, \text{m/s}^2$, $p = 1000 \, \text{kg/m}^3$).

Answer 1

The fractional compression $\frac{\Delta V}{V}$ is given by:

$\frac{\Delta V}{V} = -\frac{\Delta P}{B}$

The pressure $\Delta P$ at the bottom of the ocean is:

$\Delta P = \rho gh = 1000 \times 10 \times 4000 = 4 \times 10^7 \, \text{Pa}$

Thus,

$\frac{\Delta V}{V} = -\frac{4 \times 10^7}{2 \times 10^9} = -2 \times 10^{-2}$

Therefore, $\alpha = 2$.