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Elastic Modulie

Question

The density of a metal at normal pressure is $ρ$ . Its density when it is subjected to an excess pressure $P$ is $ρ'$ . If B is Bulk modulus of the metal, the ratio of $\frac{ρ'}{ρ}$ is :

Difficult

Solution

$\begin{array}{l} B = \frac{P}{(- \frac{Δ V}{V_{1}})}, ρ = \frac{m}{V_{1}}, ρ^{'} = \frac{m}{V_{2}} \\ \frac{Δ V}{V_{1}} = - \frac{P}{B}, \frac{Δ V}{V_{1}} = \frac{V_{2} - V_{1}}{V_{1}} = \frac{m (\frac{1}{ρ'} - \frac{1}{ρ})}{\frac{m}{ρ}} = \frac{ρ - ρ'}{ρ'} \\ - \frac{P}{B} = \frac{ρ}{ρ'} - 1 \\ 1 - \frac{P}{B} = \frac{ρ}{ρ'} \\ \Rightarrow \frac{ρ'}{ρ} = \frac{1}{1 - \frac{P}{B}} \end{array}$

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