Density and pressure

Why density is mass packed into one unit of volume and pressure is force spread over one unit of area, how to rearrange both formulas by sense rather than by triangle, and why a mixture's density is never the average of the two densities.

Ratio, proportion & rates of change · Compound units & rates

Density and pressure

Why it works

Density and pressure are both rates, exactly like speed — just with different ingredients.

Density is how much mass is packed into one unit of volume:

\text{density} = \frac{\text{mass}}{\text{volume}} \qquad (\text{g/cm}^3 \text{ or kg/m}^3).

Aluminium's density of

2.7

g/cm³ says: every single cm³ of aluminium has mass

2.7

g. That reading does the rearranging for you —

250

cm³ must have mass

250 \times 2.7 = 675

g (multiply, because each cm³ contributes), and a

579

g gold bar at

19.3

g/cm³ occupies

\frac{579}{19.3} = 30

cm³ (divide, to count how many "units of

19.3

g" fit). Sense-check instead of trusting a memorised triangle: mass should be the biggest number of the three for any density above 1.

Pressure is force spread over one unit of area:

\text{pressure} = \frac{\text{force}}{\text{area}} \qquad (\text{N/m}^2).

The same force pressed onto a smaller area gives a larger pressure — that's why a drawing pin goes in point-first. Halve the area and the pressure doubles: pressure is inversely proportional to area for a fixed force.

A mixture's density is total mass over total volume. Mix

150

cm³ of a metal with density

4.2

g/cm³ and

100

cm³ of one with density

6.7

g/cm³, and the alloy's density is not the average

5.45

. Work with the totals: the masses are

150 \times 4.2 = 630

g and

100 \times 6.7 = 670

g, so

\text{density} = \frac{630 + 670}{150 + 100} = \frac{1300}{250} = 5.2 \text{ g/cm}^3.

The plain average is only right when the two volumes are equal — otherwise the ingredient with more volume drags the density towards itself, exactly like the slow stage of a journey drags down average speed.

Units must agree before the formula fires. Grams pair with cm³; kilograms pair with m³. Mass in kg with volume in cm³ gives a meaningless number — convert first.