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)
B
(2MgAB)
C
(Mg3AB)
D
(Mg2AB)

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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