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Question

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

A
MgAB
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B
Mg4AB
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C
Mg3AB
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D
Mg2AB
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Solution

The correct option is C Mg3AB
Pressure increased due to weight M
P=ForceArea=MgA.... (1)


And we know,
ΔVV=PB
ΔVV=MgAB...... (2) [P=MgA]
where ΔVV= Fractional decrease in the volume
P= Increased Pressure
B= Bulk modulus
Volume of a sphere is V=43πR3 (where R is radius)
ΔVV=3ΔRR

Now, using (2), we have
ΔRR=Mg3AB (-ve sign indicates decrease in radius)
Fractional decrease in radius is Mg3AB

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