First slide

Elastic behaviour of solids

Question

A material has normal density $ρ$ and bulk modulus K. The increase in the density of the material when it is subjected to an external pressure P from all sides is

Moderate

A

$P / ρK$
B

$K / ρP$
C

$ρP / K$
D

$ρK / P$

Solution

$\frac{ρ^{'}}{ρ} = \frac{\frac{M}{V - ΔV}}{M / V} = \frac{V}{V - ΔV}$
$or \frac{ρ^{'}}{ρ} = V (V - ΔV)^{- 1} or \frac{ρ^{'}}{ρ} = 1 + \frac{ΔV}{V}$
(using binomial theorem)
$or \frac{ΔV}{V} = \frac{ρ^{'}}{ρ} - 1 or \frac{ΔV}{V} = \frac{ρ^{'} - ρ}{ρ}$
$Again, k = \frac{ΔP}{\frac{ΔV}{V}} = \frac{ρΔP}{ρ^{'} - ρ} or ρ^{'} - ρ = \frac{ρΔ P}{K}$
$But, ΔP = P (given)$
$∴ ρ^{'} - ρ = \frac{ρP}{K}$

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