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. $\frac{mg}{4AB}$
• B. $\frac{mg}{3AB}$
• C. $\frac{mg}{2AB}$
• D. $\frac{mg}{AB}$

Answer 1

Answer: (B)

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