A solid sphere of radius r made of a soft mat | vedclass

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(B) The bulk modulus $K$ is defined as the ratio of volumetric stress to volumetric strain: $K = \frac{\Delta P}{\left( \frac{dV}{V} \right)}$.The pre

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$\frac{mg}{3Ka}$

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(B) The bulk modulus $K$ is defined as the ratio of volumetric stress to volumetric strain: $K = \frac{\Delta P}{\left( \frac{dV}{V} \right)}$.The pressure increase $\Delta P$ due to the mass $m$ on the piston of area $a$ is $\Delta P = \frac{mg}{a}$.Substituting this into the bulk modulus formula: $K = \frac{mg/a}{dV/V} \Rightarrow \frac{dV}{V} = \frac{mg}{Ka} \dots (i)$.The volume of a sphere is $V = \frac{4}{3}\pi r^3$. Taking the logarithmic derivative,we get $\frac{dV}{V} = 3 \frac{dr}{r} \dots (ii)$.Equating $(i)$ and $(ii)$: $3 \frac{dr}{r} = \frac{mg}{Ka}$.Therefore,the fractional decrement in the radius is $\frac{dr}{r} = \frac{mg}{3Ka}$.

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