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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 the entire cross section of the 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?

mg/3Ka

mg/Ka

Ka/mg

Ka/3mg

Solution:

Volume of sphere V = (4π/3)r³
Decrease in volume of sphere -dV = 4πr²dr
Bulk modulus K = P(V/-dV) => -dV = PV/K (1)
Pressure P = Force/Area = mg/a. (2)
Using (2) in (1), we get
4πr²dr = (mg/a)(4πr³/3K)
=> dr/r = mg/3Ka