Question

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) \(\frac{Ka}{3mg}\)

Answer 1

(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),

\(\frac{dr}{r}=\frac{mg}{3Ka}\)