First slide

Elastic behaviour of solids

Question

The density of water at the surface of ocean is $ρ$ . If the bulk modulus of water is B, then the density of ocean water at depth, when the pressure is $α p_{0}$ where $p_{0}$ is the atmospheric pressure, is

Moderate

A

$\frac{ρ B}{B - (α - 1) p_{0}}$
B

$\frac{ρ B}{B + (α - 1) p_{0}}$
C

$\frac{ρ B}{B - α p_{0}}$
D

$\frac{ρ B}{B + α p_{0}}$

Solution

From the definition of bulk modulus
As we move from surface to place where pressure changes to $α p_{0}$ , let us assume volume changes by $Δ$ V, then
$B = \frac{V Δ p}{Δ V} = \frac{V (α - 1) p_{0}}{Δ V}$
$New volume, V' = V - Δ V = V [1 - \frac{(α - 1) p_{0}}{B}]$
$Density at the given depth, ρ' = ρ \frac{V}{V'}, where ρ is density at surface$
$\Rightarrow ρ' = \frac{ρ \times B}{B - (α - 1) p_{0}}$

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