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Question

# A solid sphere of radius R and made of a material of bulk modulus K is completely immersed in a liquid in a cylindrical container. A massless piston of area A floats on the surface of the liquid. When a mass M is placed on the piston to compress the liquid, the fractional change in the radius of the sphere, δRR is given by

A

MgKA

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B

Mg2KA

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C

Mg3KA

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D

Mg4KA

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Solution

## The correct option is C Mg3KA Pressure exerted by the piston on the liquid when a mass M is placed on the piston, P=MgA This pressure is exerted by the liquid equally in all directions. Therefore, the surface of the sphere experiences a force P per unit area. The stress on the sphere is P=MgA. Now, the volume of the sphere is V=4πR33 Due to stress, the change in the volume of the sphere is δV=δ(4πR33)=4π3.3R2δR=4πR2δR ∴ Volume strain δVV=3δRR By definition, bulk modulus, K=Volume StressVolume Strain=MgA3δRR ⇒δRR=Mg3KA Hence, the correct choice is (c).

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