Mechanical Properties of Solids - Result Question 9
10. The bulk modulus of a spherical object is ’ $B$ ‘. If it is subjected to uniform pressure ’ $p$ ‘, the fractional decrease in radius is
[2017]
(a) $\frac{B}{3 p}$
(b) $\frac{3 p}{B}$
(c) $\frac{p}{3 B}$
(d) $\frac{p}{B}$
Show Answer
Solution:
- (c) Bulk modulus is given by
$B=\frac{p}{(\frac{\Delta V}{V})} \quad$ or $\quad \frac{\Delta V}{V}=\frac{p}{B}$
$3 \frac{\Delta R}{R}=\frac{p}{B}$ (here, $\frac{\Delta R}{R}=$ fractional decreases in radius) $\Rightarrow \frac{\Delta R}{R}=\frac{p}{3 B}$
If the object of mass $m$ and radius $r$ is sorrounded by a liquid in a cylindrical container
$\frac{\Delta R}{R}=\frac{1}{3} \frac{\Delta V}{V}$
Bulk modulus, $K=-V \frac{\Delta P}{\Delta V}$
$ \begin{aligned} & \therefore \frac{\Delta V}{V}=\frac{\Delta P}{K}=\frac{m g}{A K} \quad(\because \Delta P=\frac{m g}{A}) \\ & \therefore \frac{\Delta R}{R}=\frac{1}{3} \frac{m g}{A K} \quad(\text{ Here } A=\text{ area of object }) \end{aligned} $
