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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 pistion of area 'A' floats on the surface of the liquid. Find the fractional change in the radius of the sphere $$(\dfrac{dR}{R})$$, when a mass M is placed on the piston to compress the liquid.


A
Mg2AB
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B
Mg3AB
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C
2MgAB
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D
3MgAB
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Solution

The correct option is B $$\dfrac{Mg}{3 AB}$$
Radius of sphere, $$R$$ Mass placed on massless piston = $$M$$ Area of piston = $$A$$
* Volume of sphere, $$V=\frac{4}{3} \pi R^{3}$$ diff wrt $$R$$ both side $$\begin{aligned} \therefore \quad-d V &=4 \pi R^{2} d R \\ \text { Bulk Moduls, } B &=\frac{d P}{-\frac{d V}{V}} \end{aligned}$$

$$\begin{aligned}B &=\frac{M g}{A} \\B &=\frac{M g}{-\frac{4 \pi R^{2} d R}{\frac{4}{3} \pi R^{3}}} \\&-\frac{d R}{R}=\frac{M g}{3 A B}\end{aligned}$$

$$\begin{array}{l}\text { Fractional decrease in radius of sphere } \\\text { is } \frac{M g}{3 R B}\end{array}$$

Physics

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