Question

The bulk modulus of $n$ spherical objects is $B$. If it is subjected to uniform pressure $P$, the fractional decrease in radius is -

• A. $\frac{P}{B}$
• B. $\frac{B}{3P}$
• C. $\frac{3P}{B}$
• D. $\frac{P}{3B}$

Answer 1

The correct option is D $\frac{P}{3B}$

We know that
Bulk modulus $=\frac{\text{Pressure applied}}{\text{Fractional change in volume}}$
or $B=\frac{P}{-\frac{\Delta V}{V}}$
$-ve$ sign shows decrease in volume
$\Rightarrow \frac{\Delta V}{V}=\frac{-P}{B}...\left(1\right)$
Also, volume of a sphere
$V=\frac{4}{3}\pi r^3$
$\Rightarrow \frac{\Delta V}{\Delta r}=\frac{4}{3}\pi .3r^2$
$\Rightarrow \frac{\Delta V}{\Delta r}=\frac{4}{3}\pi r^3\times \frac{3}{r}$
$\Rightarrow \frac{\Delta V}{\Delta r}=\frac{3V}{r}$
$\Rightarrow \frac{\Delta V}{V}=\frac{3\Delta r}{r}$
$\Rightarrow \frac{-P}{B}=\frac{3\Delta r}{r}\left[\text{from Eq.}\right(1\left)\right]$
$\therefore \frac{\Delta r}{r}=\frac{-P}{3B}$
Hence, fractional decrease in radius is $\frac{P}{3B}$.