Properties of Matter 1 Question 18
18. A solid sphere of radius $R$ made of a 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. When a mass $M$ is placed on the piston to compress the liquid the fractional change in the radius of the sphere, $\delta R / R$, is ………. .
$(1988,2 M)$
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Answer:
Correct Answer: 18. $\frac{M g}{3 A K}$
Solution:
- $\Delta p=\frac{M g}{A} \Rightarrow\left|\frac{\Delta V}{V}\right|=\frac{\Delta p}{k}=\frac{M g}{A k}$
Now as,
$$ V=\frac{4}{3} \pi R^{3} \text { or } V \propto R^{3} $$
$\therefore \quad \frac{\Delta V}{V}=3\left(\frac{\Delta R}{R}\right)$
$$ \text { or } \quad \frac{\Delta R}{R}=\frac{1}{3}\left(\frac{\Delta V}{V}\right) \cdots(i) $$
$\therefore$ From Eq. (i)
$$ \frac{\Delta R}{R}=\frac{M g}{3 A k} $$
