Question

A solid sphere of radius R made of material of bulk modulus K is surrounded by 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 $\delta R/R$ is

• A. mg/AK
• B. mg/3AK
• C. mg/A
• D. mg/3AR

Answer 1

Answer: (B)

mg/3AK

Answer 2

The additional pressure applied on the liquid surface by the mass m on the piston is $P = \frac{mg}{A}$. This pressure is transmitted equally throughout the liquid and to the sphere, according to Pascal's law. The bulk modulus (K) of a material is defined as the ratio of applied pressure (P) to the fractional decrease in volume ($\Delta V/V$): $K = - \frac{P}{\Delta V/V}$. For a sphere, the volume is $V = \frac{4}{3}\pi R^3$. The fractional change in volume is related to the fractional change in radius by: $\frac{\Delta V}{V} = 3 \frac{\Delta R}{R}$. Substituting this into the bulk modulus formula: $K = - \frac{P}{3 \frac{\Delta R}{R}}$. Rearranging to find the fractional change in radius: $\frac{\Delta R}{R} = - \frac{P}{3K}$. Substituting $P = \frac{mg}{A}$: $\frac{\Delta R}{R} = - \frac{mg/A}{3K} = - \frac{mg}{3AK}$. The fractional change $\delta R/R$ is $\frac{mg}{3AK}$.