The special case tool

Our equations in physics often have multiple symbols. You can often help to make sense of the relationships in an equation by considering special cases, in particular, extreme or limiting cases of one or more of the symbols. The icon we've chosen for consideration of extreme cases is the calipers since it measures the extremes of an object. 

To see how this works, let's consider two examples.

Example 1: An extreme case

A torus is the mathematical name for the shape of a donut or bagel. Its volume can be expressed as a function of the inner and outer radii of the torus (distance from the center to the inner and outer edge). Which of the following equations could be the correct equation for the volume of the torus?

1. $V = (2\pi R)(\pi r^2)$
2. $V = \frac{\pi^2}{4}(R+r)(R-r)^2$
3. $V = \pi Rr$
4. $V = \frac{\pi^2}{4}(Rr)^2$

You could look this up on Google, but on a test we might not have access to it. Let's see whether we could figure it out.

The first thing to try is dimensional analysis. We're asked for an equation for a volume, so it has to have the dimensions of length cubed, that is,  [V ] = L³, a the product of three lengths.

Since both r and R have dimensions of length, our equation must have a product of three of them. Choices 1 and 2 have the right dimensions, but 3 has dimensions of L² and 4 has dimensions of L⁴, so we can rule out both of those on dimensional grounds.

How to choose between choices 1 and 2? We have a 50-50 chance if we guess, but if we look at what the variables represent we might get a better idea. $R$ is the outer radius and $r$ is the inner. What happens if $r$ and $R$ are equal? As $r$  gets closer and closer to $R$, the torus gets skinnier and skinnier, going from a bagel to a bicycle wheel, to a wire ring. Clearly the volume gets smaller and smaller and has to be 0 if $r = R$. This is only true for answer 2 so we can rule out 1, and, indeed, 2 is the correct answer.

Our second example is not a limiting or extreme case, but one where knowing something about a particular example helps figure things out. 

Example 2: A special value but not an extreme case

You fly east in an airplane for 100 km. You then turn north by 60 degrees and fly 200 km. About how far are you from your starting point?

1. 170 km
2. 200 km
3. 260 km
4. 300 km
5. 370 km 

This one seems different from Example 1. There isn't even any equation! Looking at this (and if you know a bit about how to add vectors) you might expect that this should be done by using trig and taking components. Or maybe by drawing the final distance, setting a scale and measuring with a ruler. But we can do this much more easily by considering some special cases. 

First, consider if the angle were not 60^o but 0^o. In that case, the plane would be flying in a straight line so the distances would just add. We would be 300 km from our starting point after flying the two legs. As we increase the angle, we're now flying the same distance, but now along 2 legs of a triangle that add to 300 km. The third leg has to be less than that since a straight line is the shortest distance between two points. So we can rule out answers 4 and 5. 

Second, to compare the distance we are looking for to 200 km, swing the 200 km line around the destination point until it reaches along the dotted path. We can clearly see that the straight dotted path (total distance) is longer than this leg. So the answer can't be 200 km or less. That rules out answers 1 and 2. The only answer left is 3 and indeed, 3 is the correct answer.

Why these examples help

These two examples show two different ways of considering special cases. In both, we had to match the geometry of the physical example to mathematical ways of describing them.

In the first example, we had an equation. After eliminating two dimensionally non-viable answers, the trick was to figure out what special case might be relevant. The extreme case when $r = R$ allowed us to get a clear result.

In the second example, we had a geometrical situation but no equation. Our approach was to consider alternative paths, leading to special cases where we clearly knew the answer. Essentially, what we did was to make the angles of the paths into variables and see what would happen if we changed them to simpler situations. We then used some basic facts about geometry to get a clean answer.

This is illustrative of the fact that often, examples with some particular symmetry is useful and easy to work with.

Finding special or extreme cases will often help you solve a problem. Picking a special case where you know the answer can show you if you made an algebraic error.

Joe Redish 7/3/17