What If I Double It?
By Thomas Humphrey
“Three inches is such a wretched height to be,” said Alice. “It is a very good height indeed!” said the Caterpillar angrily, rearing itself upright as it spoke (it was exactly three inches high).
Alice’s Adventures in Wonderland, Lewis Carroll
Many of the things in our world are the way they are simply because of their size. Outside of Wonderland, neither Alice nor the Caterpillar could be anything but their normal, everyday sizes.
If a caterpillar were scaled up, it would collapse under its own weight; if Alice were scaled down, she'd have to spend most of her day eating to provide fuel for a body so small that it loses heat with every heartbeat.
Why does size matter?
To find out more, let's cook a turkey.
Suppose you are responsible for cooking a Thanksgiving turkey. You have a 20-pound turkey, but your cookbook only tells you how long it takes to cook a 10-pound turkey.
How long do you cook your turkey?
Since the 20-pound turkey is twice the size of a 10-pound bird, at first the answer might seem obvious: Simply double the cooking time suggested for a 10-pound turkey. But is that really the right thing to do?
The way I see it, there are three ways to answer this question:
A) You can call and ask your grandmother.
B) You can get a new cookbook.
C) You can thumb through your physics books for the turkey equation.
I began by gathering cookbook data. Betty Crocker's turkey roasting chart says that when you double the weight of a turkey, you don't have to double the cooking time. You only have to increase it from 2.9 hours for the small bird to 4.5 hours for the big one. So even though the 20-pound turkey is twice the weight of the 10-pound turkey, you only have to cook it about 1.55 times as long. Why would that be?
Let's take a more detailed look at our question. What is the "it" that we are doubling? What kind of "its" does a turkey have?
The turkey has a width, a surface area, a volume, and a weight. It has a density, a heat conductivity (how well it transfers the oven's heat into its interior), and a heat capacity (how much heat it needs to climb one degree Celsius in temperature). A turkey has a lot of "its". How do some of these factors change in going from a 10-pound turkey to a 20-pound turkey?
I often buy ButterBall-brand turkeys, but here let's imagine that I have a “ButterCube” turkey—that is, my turkey is shaped like a cube. This will make it easier to see how the various factors change.
A turkey with twice the volume (represented by cubes) does not have twice the surface area or twice the width.
Take a look at the cubical turkeys above. Try to figure out how the weight, surface area, and width differ. If you count the number of small cubes in the 10-pound turkey, you will find that there are 4 x 4 x 4, or 64 cubes. The number of cubes in the 20-pound turkey is 5 x 5 x 5, or 125 cubes. That's not exactly double, but it's pretty close. So now we know that the 20-pound ButterCube is about twice the volume of the 10-pound ButterCube (that is, it has twice as many little cubes), and therefore it weighs about twice as much.
But when you double the size of a turkey, what happens to its width and surface area? Do they double, too?
If you look at the ButterCube turkeys above, you can see that the widths of the two turkeys are 4 and 5 blocks respectively. So the bigger turkey is about 25 percent wider than the smaller one. It did not double.
If we look at surface area, the small turkey is 6 sides x 16 blocks per side, or 96 blocks. The surface area of the big turkey is 6 sides x 25 blocks per side, or 150 blocks. That means the big turkey has about 50 percent more blocks in it than the small turkey. So that measurement didn't double, either. More precisely, the width and all the other linear dimensions increased by a factor of 1.26 and the surface area increased by a factor of 1.59.
How do some of these “its”—weight, surface area, and thickness—influence the turkey’s cooking time?
Increase in cooking time
If you put the three factors together, the cooking time increases by 2 x 0.63 x 1.26 = 1.59. So the new cooking time is 2.9 hours x 1.59 = 4.6 hours. Betty says to increase the time to 4.5 hours, or by a factor of 1.55. (4.5 hr/ 2.9 hr = 1.55.) That’s pretty close!
Well, first of all, the 20-pound turkey, because it has doubled in volume, has twice as much stuff to heat up, so we need to put twice as much heat into it. Fair enough. How does the heat get in? It is transferred across the surface of the turkey, and it must travel all the way into the center of the bird. The bigger turkey has more surface. That should speed up the transfer of heat, but the heat must travel a longer way to the center. That will slow things down. The graph below shows the cumulative effect of these three factors: weight, surface area, and distance to the center.
This graph shows the impact on cooking time as a result of changing the weight, surface area, and width of a turkey.
As you can see from the graph, the net result is that it doesn't take twice as long to cook the twice-as-heavy turkey. The physicists agree with the home economist.
So now we know how to cook a turkey. But in this little foray into the physics of cooking, we discovered that the seemingly innocuous question, “What happens if you double it?” has turned out to be quite complex. We must be very specific about which feature of the turkey we are doubling because we don't seem to be able to double everything at once!
The fact that we cannot double every feature of the turkey at the same time is one example of a very general behavior in nature, a behavior that leads to consequences even more important than the difference between a perfect Thanksgiving turkey and an overcooked one. When we compare similar objects, one large and one small, not all features of the object are magnified or reduced by the same ratio. This has dramatic consequences for natural behavior, including how things move in and out of biological cells.