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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 $$ \left( \frac { d R }{ R }  \right)  $$ 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 $$ \left( \dfrac { Mg }{ 3AB } \right) $$

Given,

Radius of sphere, $$R$$

Mass placed on massless piston, $$M$$.

 Area of piston, $$A$$

Change in pressure $$\Delta P=\dfrac{\Delta F}{A}=\dfrac{Mg-0}{A}=\dfrac{Mg}{A}$$

Volume of sphere, $$v=\dfrac{4}{3}\pi {{R}^{3}}$$

Small decrease in volume, $$-dv=d\left( \dfrac{4}{3}\pi {{R}^{^{3}}} \right)=4\pi {{R}^{2}}dR$$

Bulk modulus, $$B$$

  $$ B=\dfrac{dp}{-\dfrac{dv}{v}}=\dfrac{\dfrac{Mg}{A}}{-\dfrac{4\pi {{R}^{2}}dR}{\dfrac{4}{3}\pi {{R}^{3}}}}=\dfrac{Mg}{-3A\dfrac{dR}{R}} $$

 $$ -\dfrac{dR}{R}=\dfrac{Mg}{3AB} $$

Hence, fractional decrease in radius of sphere is $$\dfrac{Mg}{3AB}$$ 


Physics

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