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 piston of area $A$ floats on the surface of the liquid. Find the fractional decrease in the radius of the sphere $\left(\frac{\Delta R}{R}\right)$, when a mass $M$ is placed on the piston to compress the liquid.

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

Answer 1

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

Pressure increased due to weight $M$
$P=\frac{\text{Force}}{\text{Area}}=\frac{Mg}{A}$.... $\left(1\right)$


And we know,
$\frac{-\Delta V}{V}=\frac{P}{B}$
$\Rightarrow \frac{-\Delta V}{V}=\frac{Mg}{AB}$...... $\left(2\right)$ $[\because P=\frac{Mg}{A}]$
where $\frac{-\Delta V}{V}=$ Fractional decrease in the volume
$P=$ Increased Pressure
$B=$ Bulk modulus
Volume of a sphere is $V=\frac{4}{3}\pi R^3$ (where $R$ is radius)
$\Rightarrow \frac{\Delta V}{V}=\frac{3\Delta R}{R}$

Now, using $\left(2\right)$, we have
$-\frac{\Delta R}{R}=\frac{Mg}{3AB}$ (-ve sign indicates decrease in radius)
$\therefore$ Fractional decrease in radius is $\frac{Mg}{3AB}$