Question

A material has normal density $\rho$ and bulk modulus $B$. The increase in the density of the material, when it is subjected to an external pressure $P$ from all sides is :

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

Answer 1

The correct option is C $\frac{P\rho }{B}$

Solution

We know that,

B (Bulk modulus) = $\frac{-P}{\frac{\Delta V}{V}}$ ___ (1)

Also, $\rho =\frac{m}{V}$

$\Rightarrow \rho =m{V}^{-1}$

$\Rightarrow \frac{\Delta \rho }{\rho }=\frac{\Delta m}{m}-\frac{\Delta V}{V}$

As mass is constant, $\Delta m=0$

So, $\frac{\Delta \rho }{\rho }=\frac{-\Delta V}{V}$ ___ (ii)

From (i) and (ii), we get

$B=\frac{-P}{\frac{-\Delta \rho }{\rho }}$

$\Rightarrow \frac{\Delta \rho }{\rho }=\frac{P}{B}$

$\Rightarrow \Delta \rho =\frac{P}{B}\rho$

Option - C is correct