#### The Scientist’s Essay for 2. Heavy for Size

... density provides a way of saying that aluminum is "heavier for size" than oak, but less so than copper, without having to specify which pieces of aluminum, oak, or copper we're talking about.

Why does a small pebble sink in water, while a much heavier piece of wood floats? Why does a balloon filled with helium float away, while one filled with air falls? Why is it hotter in the balcony than in the orchestra? How can you tell whether a piece of jewelry is really gold, or just gold-plated, without damaging it? These questions can't be answered by considering either weight or volume in isolation, but somehow involve the connection between them.

All of us have the intuitive sense that some materials are intrinsically “heavier” than others. We say that styrofoam is “light” and steel is “heavy”, even though it's perfectly possible to have a (big) piece of styrofoam that weighs more than a (small) piece of steel. What we mean is that a piece of steel is heavier for its size than a piece of styrofoam. We can nail down the rough idea of “heaviness for size” more precisely by inventing a new composite quantity, density, defined mathematically as the ratio of weight (or mass) to volume: Density = Weight/Volume. We often read this in words as “density is equal to weight per unit of volume.”

Apart from allowing us to answer questions like those in the first paragraph, the idea of density is productive as an example of what scientists call an intensive quantity. Many of the measurable quantities of a system increase in proportion to the size of the system: the size of a farmer's harvest is proportional to the acreage of her fields; the amount of gas your car burns is proportional to how far you drive; the amount of money a worker makes is proportional to the number of hours he works; the number of teachers in a school is (roughly) proportional to the number of students. These are “extensive” quantities. But for many purposes it's useful to devise ratios that are (at least roughly) independent of size: bushels of corn per acre, miles per gallon of gas, dollars per hour, number of students per teacher. You could think of lots of others: per capita income, 20% discounts, unit prices at the supermarket.

Among the virtues of intensive quantities is that they allow a “fair comparison” of things of dissimilar size — a large bag of potatoes for \$4.99, or a smaller one for \$1.99? In the case of materials, density provides a way of saying that aluminum is “heavier for size” than oak, but less so than copper, without having to specify which pieces of aluminum, oak, or copper we're talking about.

Density also provides a natural link to important and closely related ideas in mathematics, such as ratios, proportion, graphs, and linear functions. If two objects are made of the same material, for example, there is a proportional relationship between their volumes and their weights. If you plot the weights and volumes of several aluminum objects on a graph, the points will fall on a straight line. A similar graph for pieces of copper will also fall on a straight line, but a steeper one.

—Roger Tobin