Properties of Matter 1 Question 8
8. 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, $\left(\frac{d r}{r}\right)$ is
(a) $\frac{m g}{K a}$
(b) $\frac{K a}{m g}$
(c) $\frac{K a}{3 m g}$
(d) $\frac{m g}{3 K a}$
(2018 Main)
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Answer:
Correct Answer: 8. (d)
Solution:
$$ \Delta P=\frac{F}{A}=\frac{m g}{a} $$
Bulk modulus, $K=\frac{-\Delta P}{\Delta V / V}$
Here, $\quad V=\frac{4}{3} \pi r^{3}$
and $\Delta V$ or $\quad d V=\left(4 \pi r^{3}\right) d r$
$\Rightarrow \quad K=-\frac{\frac{m g}{A}}{\frac{4 \pi r^{2} d r}{\frac{4}{3} \pi r^{3}}}$ or $\frac{d r}{r}=-\frac{m g}{3 K a}$
