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, \(\left(\frac{dr}{r}\right)\) is:
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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The extra pressure is \(\frac{mg}{a}\); use \(K=\frac{\Delta P}{-\Delta V/V}\) and \(\frac{\Delta V}{V}=3\frac{\Delta r}{r}\).
Updated On: Nov 11, 2026
- \(\frac{mg}{Ka}\)
- \(\frac{Ka}{mg}\)
- \(\frac{Ka}{3mg}\)
- \(\frac{mg}{3Ka}\)
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The Correct Option is D
Solution and Explanation
Extra pressure transmitted through the liquid (Pascal's law): \(\Delta P=\frac{mg}{a}\), acting uniformly on the sphere.
Bulk modulus: \(K=\frac{\Delta P}{-\frac{\Delta V}{V}} \Rightarrow \frac{|\Delta V|}{V}=\frac{mg}{Ka}\).
For a sphere \(V=\frac43\pi r^3\), so \(\frac{dV}{V}=3\frac{dr}{r}\).
\(3\frac{dr}{r}=\frac{mg}{Ka} \Rightarrow \frac{dr}{r}=\frac{mg}{3Ka}\).
Answer: (4) \(\frac{mg}{3Ka}\)
Bulk modulus: \(K=\frac{\Delta P}{-\frac{\Delta V}{V}} \Rightarrow \frac{|\Delta V|}{V}=\frac{mg}{Ka}\).
For a sphere \(V=\frac43\pi r^3\), so \(\frac{dV}{V}=3\frac{dr}{r}\).
\(3\frac{dr}{r}=\frac{mg}{Ka} \Rightarrow \frac{dr}{r}=\frac{mg}{3Ka}\).
Answer: (4) \(\frac{mg}{3Ka}\)
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