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Section 17.1 Density

The quantity obtained by dividing mass \(M \) by volume \(V \) is called density, or more specifically, the average density. We denote density by the Greek letter rho, \(\rho \text{.}\)

\begin{equation} \rho = \frac{M}{V}.\tag{17.1.1} \end{equation}

Density has the dimensions of mass over length cubed, and has SI unit of \(\text{kg/m}^3 \text{.}\) A common unit of density is \(\text{g/cc}\text{,}\) or gram per cubic centimeter with the following conversion factor.

\begin{equation*} 1\text{ g/cc} = 1000\text{ kg/m}^3. \end{equation*}

Make sure you understand the source of the conversion factor.

If it is hard to compress a material, ie., the volume change upon compression is negligible, then the density will also not vary with compression - we say that the substance is incompressible. Compared to gases, liquids are much less compressible, and we will make the assumption that liquids are incompressible. Of course, solids are even more incompressible.

The density is a dimensionful property. Therefore, you need to keep track of units when comparing densities of two materials. For comparison purposes a more convenient dimensionless quantity, called specific gravity, is constructed by dividing the density of the material by the density of water at \(0^{\circ}\text{C}\) and one atmospheric pressure, which is \(1\text{ g/cc}\) or \(1000\text{ mg/m}^3\text{.}\)

\begin{equation*} \text{Specific gravity} = \dfrac{\text{Density of material}}{\text{Density of water}} \end{equation*}

Specific gravity, being dimensionless, provides a ready comparison among materials without having to worry about the unit of density. For instance density of aluminum is \(2.7\) in \(\text{g/cc}\) and \(2700\) in \(\text{kg/m}^3\text{,}\) but specific gravity is \(2.7\) regardless of the unit of density.