JEE Main & Advanced Physics Elasticity Fractional Change in the Radius of Sphere

A solid sphere of radius R made of a 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.

Volume of the spherical body \[V=\frac{4}{3}\pi {{R}^{3}}\]

\[\frac{\Delta V}{V}=3\frac{\Delta R}{R}\]

\[\therefore \] \[\frac{\Delta R}{R}=\frac{1}{3}\frac{\Delta V}{V}\]                          ....(i)

Bulk modulus \[K=-\,V\frac{\Delta P}{\Delta V}\]

\[\therefore \] \[\left| \frac{\Delta V}{V} \right|=\frac{\Delta P}{K}=\frac{mg}{AK}\]                     ..(ii)                                                      

\[\left[ \text{As }\Delta P=\frac{mg}{A} \right]\]

Substituting the value of \[\frac{\Delta V}{V}\] from equation (ii) in equation

(i) we get \[\frac{\Delta R}{R}=\frac{1}{3}\frac{mg}{AK}\]