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A Trip to Another Dimension!

Despite the title, we're not going to a parallel universe in this section. Instead, I'm going to talk about a useful technique called dimensional analysis.

Dimensional analysis can be used in a couple of different ways. The first way is to convert from one unit to another, say from meters per second (m/s) to miles per hour (mph). In that, it may seem boring, but can be helpful. If you have a conversion table handy, you don't really need to use dimensional analysis to change from one unit to the next. But if your table doesn't have the unit you want, you can use dimensional analysis to help use your conversion table to do the work, instead of just having to give up.

The other major way in which dimensional analysis can be used is to check whether or not your answer to a problem has the correct the units. For example, if I tell you how far it is to Disneyland from the Price Center, and tell you how long it takes me to drive to Disneyland from the Price Center, I might ask you to tell me my average speed along the drive. You probably know that the average speed is just the distance I drove, divided by the time it took me. So your answer should have units which are distance divided by time, like miles per hour. If you get hours per mile, you've got it backwards.

3.1 Dimensional analysis flavor 1: Converting units

On the CART racing circuit (the series of races for Indy cars), it is not uncommon to see the cars reach speeds of 230 miles per hour. What if I asked you to tell me how many meters an Indy car travels in one second, at a speed of 230 miles per hour? Here's how I would do it. I know that there are 1609 meters in a mile, and I know that there are 60 seconds in a minute and 60 minutes in an hour. I would write the problem down like this:


where I have used the abbreviations for miles (mi), hours (hr), meters (m), and seconds (s) and the fact that miles per hour can be written ``mi/hr''. Now, the next step is to convert miles per hour into meters per hour. To do that, I have to multiply by the number of meters in a mile, like this:


Now, I have meters per hour. What's left? I have to convert meters per hour to meters per second, which means I have to convert seconds to hours. But I don't know the number of seconds in an hour, so first, I have to convert from hours to minutes and then from minutes to seconds, like this:


So now I know that there are 3600 seconds in an hour. So the final step is to convert meters per hour to meters per second. To do that, I just multiply by 1/3600, like this:


So, the an Indy car which is cruising around the track at 230 mph is whizzing along at 102.8 m/s. That's about 112 yards in a second - longer than a football field!

The key to dimensional analysis when you are trying to convert units is that you think of the units like fractions. Instead of numbers like 1/4, though, you have things like ``m/s'' or ``km/yr'' or ``g/cm3''. And when you convert, you end up multiplying the units just as if you were multiplying fractions. That's really the key thing. Let's use our Indy car example, and do all the conversion in one step, like below:


Here, the ``mi'', ``hr'', and ``min'' all drop out, because there is one of each in the top of the units at one point and in the bottom of the units at another point; in other words, the ``numerator'' of one unit is the same as the ``denominator'' of another, and we cancel those parts of each unit. That's the key thing - we cancel tops with bottoms just like when you multiply fractions.

Here's another quick example, just to show you that this works with non-metric units just as well as with metric ones. How many 8-ounce glasses of milk can you get from a standard 1-gallon milk bottle? Well, I would go straight to a table and get the conversion from gallons to ounces, but let's say you can't. Here's another way:

1 gal × (4 quarts/1 gal) × (2 pints/1 quart) × (16 ounces/1 pint) × (1 glass/8 ounces) = 16 glasses

So you can get 16 8-ounce glasses of milk from a standard 1-gallon milk bottle.

While it's not thrilling, by any means, dimensional analysis can help you convert from one unit to another, even when your conversion tables don't have both units related to each other.

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3.2 Dimensional analysis flavor 2: Checking your answer

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 3.1, because it can help you keep from messing up - if your answer to a question has units which don't make sense, you've probably messed up somewhere.

Here's a simple equation for you:


In this equation, x is distance, v is speed, and t is time. What this equation says, of course, is that the distance you go is equal to the speed at which you are moving multiplied by the time for which you have been traveling that speed. This shouldn't sound too surprising, but how would I check to make sure that sounds right - I mean, what if the equation should really be


How might you check that? You're probably familiar with speed measured in meters per second, which can also be written as ``m/s''. As I said in Section 3.1, 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 appropriate 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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3.3 Let's sum up

Dimensional analysis is a powerful tool for anyone to have. It can help you work out whether a 5 ounce can of soup that costs $0.99 is a better deal than an 8 ounce can which costs $1.54. (It's the 8 ounce can, which costs 0.55 cents less per ounce.) It can also tell you if you have the units wrong on your homework problem, and probably therefore have done the problem incorrectly. Practicing scientists use dimensional analysis all the time in their work, as well, but really, dimensional analysis is a tool everyone ought to keep in mind.

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Greg Anderson
Tue Jan 6 16:39:23 PST 1998