Question

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

• A. $\left(\frac{3Mg}{AB}\right)$
• B. $\left(\frac{2Mg}{AB}\right)$
• C. $\left(\frac{Mg}{3AB}\right)$
• D. $\left(\frac{Mg}{2AB}\right)$

Answer 1

The correct option is C $\left(\frac{Mg}{3AB}\right)$

Given,

Radius of sphere, $R$

Mass placed on massless piston, $M$.

Area of piston, $A$

Change in pressure $\Delta P=\frac{\Delta F}{A}=\frac{Mg-0}{A}=\frac{Mg}{A}$

Volume of sphere, $v=\frac{4}{3}\pi R^3$

Small decrease in volume, $-dv=d\left(\frac{4}{3}\pi R^{{}^3}\right)=4\pi R^{2}dR$

Bulk modulus, $B$

$B=\frac{dp}{-\frac{dv}{v}}=\frac{\frac{Mg}{A}}{-\frac{4\pi R^{2}dR}{\frac{4}{3}\pi R^3}}=\frac{Mg}{-3A\frac{dR}{R}}$

$-\frac{dR}{R}=\frac{Mg}{3AB}$

Hence, fractional decrease in radius of sphere is $\frac{Mg}{3AB}$