Question

A solid sphere of radius $R$ is surrounded by a liquid in a cylindrical container. A massless piston of area $A$ floats on the surface of the liquid. Find the value of bulk modulus, if the fractional decrease in the radius of the sphere is $\kappa$ when a mass $M$ is placed on the piston to compress the liquid. [Neglect the pressure variation between the piston and sphere and consider $\kappa$ as positive]

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

Answer 1

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


When mass $M$ is placed on the piston, the pressure on the sphere increases.
Pressure increase $\left(\Delta P\right)=\frac{\text{Force}}{\text{Area}}=\frac{Mg}{A}$
and we know that
$\frac{-\Delta V}{V}=\frac{\Delta P}{B}$
where, $B=$ Bulk Modulus
and $\frac{\Delta V}{V}=$ Fractional decrease in volume

$\Rightarrow \frac{-\Delta V}{V}=\frac{Mg}{AB}$ -----(i)

Volume of sphere $\left(V\right)=\frac{4}{3}\pi R^3$
$\Rightarrow \frac{\Delta V}{V}=-3\frac{\Delta R}{R}$
where, $\frac{\Delta R}{R}=$ fractional decrement in the radius of sphere.
$\Rightarrow \frac{\Delta V}{V}=-3\kappa$
[assuming $\kappa$ is positive]

Putting the values of $\frac{\Delta V}{V}$ in eq (i)
$3\kappa =\frac{Mg}{AB}$
$\Rightarrow B=\frac{Mg}{3A\kappa }$
$\therefore B=\frac{Mg}{3A\kappa }$