A solid sphere of radius r made of a soft material of bulk modulus K… | JEE Main 2018

Q20JEE Main 2018Difficult

Density, Compressibility & Pascal's Law

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:

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Answer: (d)

Extra pressure on the liquid (Pascal's law): $P = \dfrac{mg}{a}$ , transmitted to the sphere.
$\dfrac{\Delta V}{V} = \dfrac{P}{K} = \dfrac{mg}{Ka}$
$V \propto r^{3} \Rightarrow \dfrac{\Delta V}{V} = 3\dfrac{dr}{r} \Rightarrow \dfrac{dr}{r} = \dfrac{mg}{3Ka}$
So the answer is (d) $\dfrac{mg}{3Ka}$ .

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