Question

A small solid sphere of radius R made of a material of bulk modulus B is surrounded by an incompressible liquid in a cylindrical container. A massless piston of area A floats on the surface of the liquid. The magnitude of fractional change in the radius of the sphere $\left(\frac{dR}{R}\right)$when a mass M is placed slowly on the piston to compress the liquid is:

• A. $\frac{Mg}{3AB}$
• B. $\frac{Mg}{AB}$
• C. $\frac{3Mg}{AB}$
• D. None of these

Answer 1

The correct option is A $\frac{Mg}{3AB}$

Pressure formula,
$P=P_0+\rho gh$

and $P^{\prime }=P_0+\rho gh+\frac{Mg}{A}$
So, $\Delta P=\frac{Mg}{A}$
Bulk modulus, $B=\frac{\Delta P}{\Delta V/V}$
or, $\frac{\Delta V}{V}=\frac{\Delta P}{B}$
Now, $V=\frac{4}{3}\pi r^3$
So, $lnV=ln\frac{4}{3}+ln\pi +3lnR$
or, $\frac{\Delta V}{V}=\frac{3\Delta R}{R}$
or, $\frac{\Delta R}{R}=\frac{\Delta V}{3V}=\frac{\Delta P}{3B}=\frac{Mg}{3AB}$