Question

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

• A. $P / \rho K$
• B. $K / \rho P$
• C. $\rho P / K$
• D. $\rho K / P$

Answer 1

Answer: (C)

$\frac{\rho'}{\rho}=\frac{\frac{M}{V-\Delta V}}{M / V}=\frac{V}{V-\Delta V}$
or $\frac{\rho'}{\rho}=V(V-\Delta V)^{-1}$
or $\frac{\rho'}{\rho}=1+\frac{\Delta V}{V}$ (using binomial theorem)
or $\frac{\Delta V}{V}=\frac{\rho'}{\rho}-1$
or $\frac{\Delta V}{V}=\frac{\rho'-\rho}{\rho}$
Again, $k=k=\frac{\Delta P}{\frac{\Delta V}{V}}=\frac{\rho \Delta P}{\rho'-\rho}$
or $\rho'-\rho=\frac{\rho \Delta R}{K}$
But, $\Delta P=P$ (given)
$\therefore \rho'-\rho=\frac{\rho P}{K}$