# A solid sphere of radius r made of a soft material of bulk modulus K is surrounded by a liquid in a cylindrical container.

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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, $(\frac{dr}{r})$, ,is: where negative sign indicates volume decreases with increase in pressure.

(A) $\frac{mg}{3Ka}$

(B) $\frac{mg}{Ka}$

(C) $\frac{Ka}{mg}$

(D) $\frac{Ka}{3mg}$

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(A) $\frac{mg}{3Ka}$

Bulk modulus is given as,

K = $(\frac{-dP}{dV/V})$

where negative sign indicates volume decreases with increase in pressure.

∴ Fractional decrease in volume will be,

$\frac{dV}{V}=\frac{dP}{K}$ If area of cross-section of cylinder is a, then,

$\frac{dV}{V}=\frac{dP}{K}$ If area of cross-section of cylinder is a, then,

Also,

Equating equations (i) and (ii),

$\frac{dr}{r}=\frac{mg}{3Ka}$

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