Let's talk about how you can use dimensional analysis to check your answers when you are finished with a problem. This is actually a more important use of dimensional analysis that the one described in Section 3.2, because it can help you keep from messing up -- if the units you get in answer to a question don't make sense, you've probably messed up somewhere.

Here's a simple equation for you:

How might you check that? You're probably familiar with speed measured in meters per second, which can also be written as ``m/s'' (see Section 3.2 if that's not familiar for some reason). As I said in Section 3.2, you can treat units as a fraction in dimensional analysis. So, let's use dimensional analysis on Equation 7 to prove that that equation is wrong. First, let's write it with the units one would expect for distance, speed, and time

and then let's use dimensional analysis to see if the units make sense.

So it looks like the units make no sense. Meters is not the same as meters-seconds, and who has ever heard of meters-seconds being used to measure distance? Nobody, because meters-seconds is a meaningless unit. So clearly, Equation 7 is wrong. Does Equation 6 work any better? Let's see. Again, we write out Equation 6 in terms of the units and use dimensional analysis to check them.

Yes, it works. Meters is the unit of distance, and since we get meters on both sides of the equation, Equation 6 works. So distance really is speed multiplied by time; but then, you knew that.

This is a simple example, but dimensional analysis works like this for
*any*
problem, provided you know what units you *should* get for a particular
problem. If you get those units, you're not sure the answer is right, of
course -- you may still have made a mistake in the math. But if you
*don't* get those units, the chances are very high that either (a) you
were wrong with what you should expect or (b) you got the answer really
wrong. If you check to make sure that the expected units were what you thought
they should be, then you are left with (b), and you know via dimensional
analysis that you got the answer wrong.

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Greg Anderson

ganderson@ucsd.edu

Tue Jan 14 10:38:34 PST 1997