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  • A solid sphere of radius R and made of a material of bulk modulus K is completely immersed in a liquid in a cylindrical container. A massless piston of area A floats on the surface o

A solid sphere of radius R and made of a material of bulk modulus K is completely immersed in a liquid in a cylindrical container. A massless piston of area A floats on the surface of the liquid. When a mass M is placed on the piston to compress the liquid, the fractional change in the radius of the sphere, \mathrm{\delta R / R} is given by

Option: 1

\mathrm{\frac{M g}{K A}}


Option: 2

\mathrm{\frac{M g}{2 K A}}


Option: 3

\mathrm{\frac{M g}{3 K A}}


Option: 4

\mathrm{\frac{M g}{4 K A}}


Answers (1)

Pressure exerted by the piston on the liquid when a mass M is placed on the piston, \mathrm{P=M g / A.} This pressure is exerted by the liquid equally in all directions. Therefore, the surface of the sphere experiences a force P per unit area. The stress on the sphere is \mathrm{P=M g / A.} Now, the volume of the sphere is

                                         \mathrm{ V=\frac{4 \pi R^3}{3} }
Due to stress, the change in the volume of the sphere is

                 \mathrm{ \begin{aligned} & \Delta V=\Delta\left(\frac{4 \pi R^3}{3}\right)=\frac{4 \pi}{3} \cdot 3 R^2 \Delta R \\ & =4 \pi R^2 \Delta R \\ & \therefore \quad \text { Volume strain } \frac{\Delta V}{V}=\frac{3 \Delta R}{R} \\ & \end{aligned} }
By definition, bulk modulus
                        \mathrm{ \begin{aligned} K & =\frac{\text { stress }}{\text { strain }}=\frac{M g / A}{3 \Delta R / R} \\ \text { or } \quad \frac{\Delta R}{R} & =\frac{M g}{3 K A} \end{aligned} }

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Kshitij

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