Question

A solid sphere of radius $r$ made of a soft material of bulk modulus $K$ is surrounded by a liquid in a cylindrical container. A massless piston of area $a$ float on the surface of the liquid, covering entire cross section of cylindrical container. When a mass $m$ is placed on the surface of the piston to compress the liquid, the fractional decrement in the radius of the sphere,

• A. $\frac{Ka}{3mg}$
• B. $\frac{Ka}{mg}$
• C. $\frac{mg}{Ka}$
• D. $\frac{mg}{3Ka}$

Answer 1

The correct option is D $\frac{mg}{3Ka}$

Formula used: Bulk Modulus $K=\frac{\text{volumetric stress}}{\text{volumetric strain}}$

Excess pressure applied on piston will be $mg/a$

Strain in sphere is given by $\frac{dV}{V}$

Using the formula of Bulk Modulus

$K=\frac{\frac{mg}{A}}{\left(\frac{dV}{V}\right)}$

$\frac{dV}{V}=\frac{mg}{Ka}\dots \left(i\right)$

Volume of sphere, $V=\frac{4}{3}\pi r^3$

Fractional change in volume,

$\frac{dV}{V}=\frac{3dr}{r}\dots \left(ii\right)$

Using eq. $\left(i\right)$ and $\left(ii\right)$

$\frac{dr}{r}=\frac{mg}{3Ka}$

FINAL ANSWER: (d)