First slide

Elastic behaviour of solids

Question

A solid sphere of radius R, made up of a material of bulk modulus K is surrounded by a liquid in a cylindrical container. A massless piston of area A floats on the surface of the liquid. When a mass M is placed on the piston to compress the liquid, the fractional change in the radius of the sphere is

Moderate

A

$\frac{Mg}{AK}$
B

$\frac{Mg}{3 AK}$
C

$\frac{3 Mg}{AK}$
D

$\frac{Mg}{2 AK}$

Solution

Change in pressure due to placing of mass on piston is $Δp = \frac{Mg}{A}$
From bulk modulus definition $K = \frac{- dp}{dV / V}$
$|\frac{dV}{V}| = \frac{Δp}{K} = \frac{Mg}{AK}$
From $V = \frac{4}{3} {πr}^{3}$
$\frac{dV}{V} = \frac{3 dR}{R} \Rightarrow \frac{dR}{R} = \frac{1}{3} \frac{dV}{V} = \frac{Mg}{3 AK}$

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