Question

A solid sphere of radius R and made of a material of bulk modulus K is completely immersed in a liquid in a cylindrical container. A massless piston of area A floats on the surface of the liquid. When a mass M is placed on the piston to compress the liquid, the fractional change in the radius of the sphere, $\frac{\delta R}{R}$ is given by

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

Answer 1

The correct option is C

$\frac{Mg}{3KA}$

Pressure exerted by the piston on the liquid when a mass M is placed on the piston, $P=\frac{Mg}{A}$
This pressure is exerted by the liquid equally in all directions. Therefore, the surface of the sphere experiences a force P per unit area. The stress on the sphere is $P=\frac{Mg}{A}$.
Now, the volume of the sphere is $V=\frac{4\pi R^3}{3}$
Due to stress, the change in the volume of the sphere is
$\delta V=\delta \left(\frac{4\pi R^3}{3}\right)=\frac{4\pi }{3}.3R^2\delta R=4\pi R^2\delta R$
$\therefore$ Volume strain $\frac{\delta V}{V}=\frac{3\delta R}{R}$
By definition, bulk modulus, $K=\frac{VolumeStress}{VolumeStrain}=\frac{\frac{Mg}{A}}{\frac{3\delta R}{R}}$
$\Rightarrow \frac{\delta R}{R}=\frac{Mg}{3KA}$
Hence, the correct choice is (c).