Density ($\rho$) is the mass per unit volume of a substance: $$ \begin{equation}\begin{aligned} density &= \frac{mass}{volume}\\ \rho &= \frac{m}{V}\\ \end{aligned}\end{equation} $$
The SI unit for density is thus the kilogram per cubic metre: $$ \begin{equation}\begin{aligned} \rho &= \frac{m}{V}\\ &\rightarrow \frac{kg}{m^3}\\ &= kgm^{-3}\\ \end{aligned}\end{equation} $$
Defining relative density
The relative density of the substance is the ratio of the density of the substance to the density of a standard (normally we use water): $$ \begin{equation}\begin{aligned} \rho_{rel} &= \frac{\rho_{substance}}{\rho_{standard}}\\ &= \frac{\rho_{substance}}{\rho_{water}}\\ \end{aligned}\end{equation} $$
The relative density tells us how many times a substance is denser than the standard (water).
Relative density and Archimedes’ Principle
Consider the case where we using an equal volume, $V$ of the substance and water: $$ \begin{equation}\begin{aligned} \rho_{rel} &= \frac{\rho_{substance}}{\rho_{water}}\\ &= \frac{\rho_{substance}}{\rho_{water}}\times \frac{V}{V}\\ &= \frac{m_{substance}}{m_{water}}\\ \end{aligned}\end{equation} $$
Taking the masses and multiplying by acceleration due to gravity: $$ \begin{equation}\begin{aligned} \rho_{rel} &= \frac{m_{substance}}{m_{water}} \times \frac{g}{g}\\ &= \frac{W_{substance}}{W_{water}}\\ \end{aligned}\end{equation} $$
Thus the relative density is the ratio of the weights of equal volumes of a substance and water. By Archimedes’ Principle, the weight of the (equal volume of) water being displaced by the object is equal to the upthrust: $$ \begin{equation}\begin{aligned} \rho_{rel} &= \frac{W_{substance}}{Upthrust}\\ \end{aligned}\end{equation} $$
This upthrust is the apparent loss in weight of the object as it moves from air to water: $$ \begin{equation}\begin{aligned} \rho_{rel} &= \frac{W_{substance}}{\text{apparent loss in weight}}\\ &= \frac{W_{substance(\text{in air})}}{W_{substance(\text{in air})}-W_{substance(\text{in water})}}\\ \end{aligned}\end{equation} $$
Thus if we use a spring balance to measure the weight of the substance in air and the weight of the substance when submerged in water, we will have sufficient information to find its relative density.
Will it sink or float?
The relative density can be used to determine if the substance will float when submerged in the standard:
| Density | Result |
|---|---|
| Less dense than water ($\rho \lt 1$) | Float |
| Same density as water ($\rho = 1$) | Stay suspended at the same depth it was placed (neutral buoyancy) |
| Denser than water ($\rho \gt 1$) | Sink |