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Question

# A solid sphere of radius R made of a material of bulk modulus B surrounded by a liquid in a cylindrical container.A massless piston of area A floats on the surface of the liquid. Find the fractional decreases in the radius of the sphere (dRR) when a mass M is placed on the piston to compress the liquid:

A
(3MgAB)
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B
(2MgAB)
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C
(Mg3AB)
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D
(Mg2AB)
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Solution

## The correct option is C (Mg3AB)Given, Radius of sphere, R Mass placed on massless piston, M. Area of piston, A Change in pressure ΔP=ΔFA=Mg−0A=MgA Volume of sphere, v=43πR3 Small decrease in volume, −dv=d(43πR3)=4πR2dR Bulk modulus, B B=dp−dvv=MgA−4πR2dR43πR3=Mg−3AdRR −dRR=Mg3AB Hence, fractional decrease in radius of sphere is Mg3AB

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