While trying to weigh John using a ruler, we made a leap from physics to geometry. Now, we are about to make a similar leap again, this time using density.

A horse and a marble cube

Imagine you are quarrying marble in the mountains of Italy, where Michelangelo once sourced marble for his sculptures.

You have a draft horse that you use to transport the marble. But you know that the maximum load it can carry is 500 kg, no more. Otherwise, your horse will collapse from the strain, fall, and say, “That’s it, boss, you can haul this marble yourself from here on out!”

Question: How do you weigh a huge cube of marble if your scale can only handle a maximum of, say, 5 kg?

This is where density comes to our rescue. How?

To do this, we must cut out a small cube from the marble with dimensions:

1 cm⋅1 cm⋅1 cm 1\,\text{cm} \cdot 1\,\text{cm} \cdot 1\,\text{cm}

That is, a volume of 1 cm31\,\text{cm}^3 .

Place this small cube on the scale and measure its weight. Let’s say it weighs 2.7 grams2.7\,\text{grams} !

We now know the density of our marble: it is 2.7 grams2.7\,\text{grams} per 1 cm31\,\text{cm}^3 .

In writing, density is denoted by the Greek letter ρ\rho (pronounced “rho”).

Knowing the density of marble, we can easily calculate the mass of our huge cube.

To do this, we need to calculate its volume in cubic centimeters. Let’s assume our cube measures 1 meter⋅1 meter⋅1 meter1\,\text{meter} \cdot 1\,\text{meter} \cdot 1\,\text{meter} .

1 meter=100 cm 1\,\text{meter} = 100\,\text{cm}

Therefore, the total volume of our cube is: 100⋅100⋅100=1,000,000100 \cdot 100 \cdot 100 = 1{,}000{,}000 or 1 million cm31\,\text{million}\,\text{cm}^3 .

Since we already know that 1 cm31\,\text{cm}^3 weighs 2.7 grams2.7\,\text{grams} , then one million cubic centimeters will weigh:

2,700,000 gramsor2700 kilograms 2{,}700{,}000\,\text{grams} \quad \text{or} \quad 2700\,\text{kilograms} (1 kilogram=1000 grams) (1\,\text{kilogram} = 1000\,\text{grams})

Horse: That’s it, boss, unload your cube—we agreed on only 500 kg! But you’ve got 2700!

So, if we calculate the density ρ\rho of a substance once, then next time, knowing the volume of an object, we can easily determine its mass using the formula:

m=ρ⋅V m = \rho \cdot V

This is how a physics problem transforms into a geometry problem: the ability to calculate the volume of a given body (object).

Mass Density Volume

Notes

  • The density of marble is not a constant value: it depends on the specific quarry location—and even within the same deposit, areas with varying densities can be found, requiring careful control during extraction and sorting.

  • In our calculation, we assume that the large cube is homogeneous and has the same density as the small sample.

  • If a person thinks their horse is talking to them, it’s probably time for them to take a break.

Food for Thought

Write down what density looks like from the perspective of dimensions.

Answer

MassLength⋅Length⋅Length \frac{\text{Mass}}{\text{Length} \cdot \text{Length} \cdot \text{Length}}

or

ML⋅L⋅L \frac{M}{L \cdot L \cdot L}

or

ML3 \frac{M}{L^3}