Density ($\rho$) is the mass per unit volume:
$$ \begin{equation}\begin{aligned} \rho&=\frac{mass}{volume}\\ &=\frac{m}{V}\\ \end{aligned}\end{equation} $$Thus the SI unit is:
$$ \begin{equation}\begin{aligned} \rho&=\frac{m}{V}\\ &\rightarrow \frac{kg}{m^3}\\ &=kgm^{-3}\\ \end{aligned}\end{equation} $$Example: Find the density of a material whose volume is $4m^3$ and mass is $7kg$.
Solution: $$ \begin{equation}\begin{aligned} \rho &= \frac{m}{V}\\ &=\frac{7kg}{4m^3}\\ &=1.75kgm^{-3}\\ \end{aligned}\end{equation} $$
Relative density
This is the ratio of the density of a substance relative to the density of a standard, usually water: $$ \begin{equation}\begin{aligned} \rho_{rel}&=\frac{\rho_{substance}}{\rho_{standard}}\\ \end{aligned}\end{equation} $$
Example: Find the relative density of gold if its density is $19,300 kgm^{-3}$.
Solution: $$ \begin{equation}\begin{aligned} \rho_{rel}&=\frac{19,300kgm^{-3}}{1000kgm^{-3}}\\ &=19.3\\ \end{aligned}\end{equation} $$
Notice that the relative density of a substance does not have any units. It is thus referred to as a dimensionless quantity.
Sinking and floating
If a substance or object is submerged in a fluid (a liquid or a gas), and its density is greater than that of the fluid, then the substance/object will sink in the fluid. Conversely, if the substance/object is lighter (less dense) than the fluid, it will float in the fluid:
- Helium floats because it is lighter than air
- Carbon dioxide sinks in air because it is denser than air
- Gold sinks in water because it is denser than water
- Air floats in water because it is less dense than the water
If the density of a substance/object is greater than that of a fluid, then its relative density will be greater than that of the fluid. Thus we can compare the relative densities in order to determine if sinking or floating will occur.