Question

A solid sphere of radius $R$ made of material of bulk modulus B is surrounded by a liquid in a cylindrical container. A massless piston of area A floats on the surface of a liquid when a mass m is placed on the piston to compress the liquid, the fractional change in the radius of the sphere $\frac{dR}{R}$ is

• A. $\frac{m g}{4 A B}$
• B. $\frac{m g}{3 A B}$
• C. $\frac{m g}{2 A B}$
• D. $\frac{m g}{A B}$

Answer 1

Answer: (B)

$B=\frac{d p}{\left(\frac{d V}{V}\right)}$
$\Rightarrow B=\frac{m g / A}{\frac{d V}{V}}$
$\Rightarrow \frac{d V}{V}=\frac{m g}{AB}$
For a sphere
$V=\frac{4 \pi R^{3}}{3}$
$\Rightarrow dV=\frac{4 \pi}{3} d(R)^{3}$
$dV=\frac{4 \pi}{3}\left(3 R^{2} d R\right)$
$\Rightarrow \frac{d V}{V}=\frac{3 d R}{R}$
$\frac{mg}{3AB}=\frac{dR}{R}$