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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 floats 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$\left( {\frac{{dr}}{r}} \right)$ is
$\frac{{Ka}}{{3mg}}$
$\frac{{mg}}{{3Ka}}$
$\frac{{mg}}{{ka}}$
$\frac{{Ka}}{{mg}}$
Solution
$Bulk\,modulus,\,K = \frac{{volumetric\,stress}}{{volumetric\,strain}}$
$K = \frac{{mg}}{{a\left( {\frac{{dV}}{V}} \right)}}\, \Rightarrow \frac{{dV}}{V} = \frac{{mg}}{{Ka}}\,\,\,\,\,\,\,\,\,\,\,\,\,…\left( i \right)$
$Volume\,of\,sphere,\,V = \frac{4}{3}\pi {R^3}$
$Fractional\,change\,in\,volume\frac{{dV}}{V} = \frac{{3dr}}{r}\,\,\,\,\,…\left( {ii} \right)$
Using eq. $(i)$ & $(ii)$ $\frac{{3dr}}{r} = \frac{{mg}}{{Ka}}$
$\therefore \frac{{dr}}{r} = \frac{{mg}}{{3Ka}}\left( {fractional\,decrement\,in\,radius} \right)$
