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: $\frac{M g}{A B}=0.3$ )

Answer 1

Increase in pressure is $\Delta p=\frac{M g}{A}$
Bulk modulus is $B=\frac{\Delta p}{(\Delta V / V)}$
$\therefore \frac{\Delta V}{V}=\frac{\Delta p}{B}=\frac{M g}{A B}$ ...(i)
Also, the volume of the sphere is $V=\frac{4}{3} \pi R^{3}$
$\Rightarrow \frac{\Delta V}{V}=\frac{3 \Delta R}{R}$
or $\frac{\Delta R}{R}=\frac{1}{3} \cdot \frac{\Delta V}{V}$,
using Eq. (i) we get
$\frac{\Delta R}{R}=\frac{M g}{3 A B}=\frac{0.3}{3}=0.1$