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 pistion of area 'A' floats on the surface of the liquid. Find the fractional change 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. $\frac{Mg}{2AB}$
• B. $\frac{Mg}{3AB}$
• C. $\frac{2Mg}{AB}$
• D. $\frac{3Mg}{AB}$

Answer 1

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

Radius of sphere, $R$ Mass placed on massless piston = $M$ Area of piston = $A$

* Volume of sphere, $V=\frac{4}{3}\pi R^3$ diff wrt $R$ both side $\begin{array}{cc}\therefore -dV& =4\pi R^{2}dR\\ \text{Bulk Moduls,}B& =\frac{dP}{-\frac{dV}{V}}\end{array}$

$\begin{array}{cc}B& =\frac{Mg}{A}\\ B& =\frac{Mg}{-\frac{4\pi R^{2}dR}{\frac{4}{3}\pi R^3}}\\ & -\frac{dR}{R}=\frac{Mg}{3AB}\end{array}$

$\begin{array}{c}\text{Fractional decrease in radius of sphere}\\ \text{is}\frac{Mg}{3RB}\end{array}$