Probably many of you are now thinking about how much you hated your high school physics class. What was that formula you used for falling bowling balls? Geez, who wants to remember that equation and solve it?
But let's say that we just want an approximate answer. In physics, there's a special method called dimensional analysis which is useful to find approximate answers to questions like this.
The first step is to figure out what quantities could be involved in the answer. For the bowling ball, it could only depend on the mass, the height, and the strength of gravity. Then we figure out the units of all these quantities, as well as the units of the answer.
|Mass of ball||kilograms|
|Height of ball||meters|
|Strength of gravity||meters/second2|
|Time to fall||seconds|
How can we combine the three quantities to get the answer? One thing's for sure, they'd have to be combined in a way that gives the correct units. So here's a guess:
Math can be hard. But in the end, the math just ends up with a number, like the square root of two. Or maybe it ends up with one half. Or 2*pi. Or something. It's highly unlikely that the math will end up with a factor of a thousand. So whatever you get, you're probably at least within a factor of 10.
I've seen one exception where dimensional analysis is off by more than a factor of 10. My last physics post discussed energy lost by radiation from a thermal pot. According to the Stefan-Boltzmann law, the rate of energy loss per unit surface area is proportional to temperature to the fourth power. What I didn't tell you is that the Stefan-Boltzmann law can almost be derived by dimensional analysis.
The first step is to figure out what quantities could be involved in the answer. To do this, you need to know something about the physics, but not much. It turns out that the answer depends only on the temperature and fundamental constants.
|Speed of light||c||meters/second|
|Planck's constant||h||Joule seconds|
|Power loss per unit area||P/A||Joules/(second meter2)|
If you try dimensional analysis on this problem, you predict the following:
Riemann zeta function).
But the dimensional analysis wasn't a complete failure. We at least showed that the power loss is proportional to temperature to the fourth power. That's the most important result!
I have one last example of dimensional analysis used in particle physics. One of the holy grails in particle physics is to figure out the correct way to combine General Relativity and Quantum Field Theory. Now, nobody knows for sure the underlying physics of the situation. However, we do know that we can combine fundamental constants to get a special length, called the Planck length.
The significance of the Planck length depends on what theory we're using. String theorists think that the Planck length is about the length of a string. If the universe has extra dimensions, perhaps these extra dimensions are Planck length in size. Some theorists think that space is quantized into lengths about the Planck length.
And since nobody really knows the underlying physics, nobody knows what mathematical factors may appear. Maybe a factor of pi will show up. At worst, a factor of 40 might appear. But even if they're off by a lot, one thing's for sure: the Planck length is tiny! If strings exist, strings will be tiny! If space is quantized, it's quantized into really tiny pieces! Dimensional analysis tells us that much.
I wonder how mathematicians and philosophers would react to the method of dimensional analysis. I suspect a lot of head-banging would be involved. How can those dang physicists be sure that this is sound reasoning? Well, no one is really sure. Luckily we can use experiments for external verification.