First slide

Elastic Modulie

Question

The normal density of gold is p and its bulk modulus is K The increase in density of a lump of gold when a pressure P applied uniformly on all side is

Moderate

A

$ρ P / K$
B

$ρK / P$
C

$P / ρK$
D

$K / ρP$

Solution

$K = \frac{P}{(ΔV / V)} or \frac{ΔV}{V} = \frac{P}{K}$ ………..(1)
Further. $ρ = \frac{M}{V} and ρ^{'} = \frac{M}{V - ΔV}$
$∴ \frac{ρ^{'}}{ρ} = \frac{V}{V - ΔV} = \frac{1}{{1 - (ΔV / V)}}$
$= {(1 - \frac{ΔV}{V})}^{- 1} = (1 + \frac{ΔV}{V})$ ………..(2)
From eqs. (1) and (2), we get
$\frac{ρ^{'}}{ρ} = (1 + \frac{P}{K}) or \frac{ρ^{'}}{ρ} - 1 = \frac{P}{K}$
or $\frac{ρ^{'} - ρ}{ρ} = \frac{ρ}{K} or ρ^{'} - ρ = \frac{ρP}{K}$

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