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A solid sphere of radius r is 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 (dr/r) is:

1. Ka/mg
2. Ka/3mg
3. mg/3Ka
4. mg/Ka

Correct answer: mg/3Ka

Solution

The correct option is derived from the relationship between pressure change and volume change in the sphere due to the applied force from the mass on the piston. The fractional decrement in the radius is proportional to the force applied (mg) divided by the bulk modulus (K) and the geometric factor related to the sphere's volume, which introduces the factor of 3 in the denominator.

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