First slide

Elastic behaviour of solids

Question

A solid sphere of radius R made of a material of bulk modulus B is surrounded by a liquid in a cylindrical container. A massless piston of area A (the area of container is also A) floats on the surface of the liquid. When a mass m is placed on the piston to compress the liquid, find the fractional change in radius of the sphere. (Given, $\frac{Mg}{AB}$ = 0.3)

Moderate

Solution

Increase in pressure is $Δp = \frac{Mg}{A}$
Bulk modulus is $Δp = \frac{Mg}{A}$
$∴ \frac{ΔV}{V} = \frac{Δp}{B} = \frac{Mg}{AB}$ …………..(i)
Also, the volume of the sphere is $V = \frac{4}{3} {πR}^{3} \Rightarrow \frac{ΔV}{V} = \frac{3 ΔR}{R}$
or $\frac{ΔR}{R} = \frac{1}{3} \cdot \frac{ΔV}{V}$ using Eq. (i) we get $\frac{ΔR}{R} = \frac{Mg}{3 AB} = \frac{0 .3}{3} = 0 .1$

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