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)

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