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Question

A solid sphere of radius r made of a soft material of bulk modulus K is surrounded by a liquid in a cylindrical container. A massless piston of area a floats on the surface of the liquid, covering entire cross section of cylindrical container. When a mass m is placed on the surface of the piston to compress the liquid, the fractional decrement in the radius of the sphere, $$\left(\displaystyle\frac{dr}{r}\right)$$, is?


A
mg3Ka
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B
mgKa
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C
Kamg
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D
Ka3mg
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Solution

The correct option is C $$\displaystyle\frac{mg}{3 Ka}$$
Volume of sphere  $$V = \dfrac{4\pi}{3}r^3$$
Decrease in volume of sphere  $$-dV = \dfrac{4\pi}{3} 3r^2 dr$$
Bulk modulus   $$K = \dfrac{P V}{-d V}$$
$$\implies \ $$   $$-dV = \dfrac{PV}{K}$$    ........(1)
Pressure   $$P = \dfrac{Force}{Area} = \dfrac{mg}{a}$$          ......(2)
Using  (2) in (1), we get    
$$\dfrac{4\pi }{3}3r^2 dr = \dfrac{mg}{a}.\dfrac{4\pi}{3}r^3.\dfrac{1}{K}$$
$$\implies \ \dfrac{dr}{r} = \dfrac{mg}{3Ka}$$

Physics

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