A solid sphere of radius $R$ made of 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
Diffcult
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Volume of the sphere $V=\frac{4 \pi}{3} R^{3}$
$\Longrightarrow \quad \frac{\delta V}{V}=\frac{3 \delta R}{R}$
Pressure exerted on the sphere by the piston $\quad P=\frac{F}{A}=\frac{M g}{A}$
Bulk modulus $\quad K=-\frac{P}{\delta V / V}$
OR $\quad K=-\frac{M g / A}{3 \delta R / R}$
$\Longrightarrow \quad \frac{\delta R}{R}=-\frac{M g}{3 A K}$
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