Question

If the bulk modulus of a body is ‘B’ and its density is $\rho$. If the increase of pressure on the body is ‘P’ then find a) new density b) change of density.

Answer 1

Hint: The bulk modulus is a fixed that describes how opposing a material is to compression. It is described as the ratio between pressure rise and the resulting reduction in a material's volume. While it implements the uniform compression of any material, it is most often utilized to define the nature of fluids.

Complete step-by-step solution:
We have given the bulk modulus of a body is ‘B’ and its density is $\rho$.
The formula for Bulk modulus is given by:
$B = \dfrac{-P}{\dfrac{\Delta V }{V}}$……($1$)
We know density is equal to the mass divided by volume.
$\rho = \dfrac{m}{V}$
$\implies \rho = mV^{-1}$
Now differentiate the above formula:
$\dfrac{\Delta \rho}{\rho} = \dfrac{\Delta m}{m} - \dfrac{\Delta V}{V}$
$\Delta m =0$
$\dfrac{\Delta \rho}{\rho} = - \dfrac{\Delta V}{V}$......($2$)
Combining equation ($1$) & ($2$):
$B = \dfrac{P}{\dfrac{\Delta \rho }{\rho}}$
$\Delta \rho = \dfrac{P \rho}{B}$
The change in density is $\dfrac{P \rho}{B}$.
Let $\rho’$ be the new density.
$\rho’ - \rho = \dfrac{P \rho}{B}$
$\rho’ = \rho \left( 1 + \dfrac{P \rho}{B} \right)$
The new density is $ \rho \left( 1 + \dfrac{P \rho}{B} \right)$.

Note:The Bulk modulus value changes depending on the nature of the matter of a sample and the temperature. In liquids, the amount of dissolved gas dramatically impacts the value. A significant value of bulk modulus means a material opposes compression, while a feeble value means volume appreciably reduces under constant pressure. The bulk modulus's inverse is compressibility, so a material with a low bulk modulus has large compressibility.