Dimensional Analysis

We've talked about this subject in the past, but I thought I would bring it up again.

While working with my son, I've been trying to get him to look at units carefully, but unfortunately, I don't get a lot of help from his Glencoe Algebra I textbook ("units" is not even in the index). It's not as if they think that units are unimportant, but they don't go into details. For most problems, they take care of themselves if you think about it a little bit, so my son doesn't like it when I try to get more formal.

For example, if they talk about 60 miles per hour, I try to get him to think of it as

60 * 1 mile / 1 hour

where the units can stand alone and are multiplied and divided just like regular numbers. I don't want him thinking that there is a units called "miles per hour". I want him to treat units like numbers, and manipulate them like numbers. The base units are "miles" and "hours"; distance and time.

This came up when we were doing some more complicated D=RT problems. When problems are simple, units are simple. When problems get complicated, then units can cause your brain to explode. I told my son that units are your friends. They can tell you if you are trying to add apples with oranges.

Therefore, I've been pushing the idea that units are just like factors that are multiplied and divided and squared. If you calculate the area of a rectangle, you might multiply 5 ft times 7 ft. I want him to look at it like this:

5*(1 ft) * 7*(1 ft)

Where the units are separate and can be manipulated like numbers. I told him that after a while, he can think of "1 ft" as simply "ft". Since everything is multiplied together, you could think of it as:

5*7 * ft*ft

or

35 *(1 ft)^2


By now, he is annoyed and wondering why this formality is necessary. I explain that if he keeps the units with the numbers at all times and multiplies and divides them just like the numbers, the units will tell him if he is making a mistake. This is hard to justify with easy problems where the units are obvious, but you don't want to wait until the problems get difficult to add in these concepts of units.

I also explained to him that if you talk about something like moment: weight times distance, you can get "ft-lbs" for the units. I told him that the dash is commonly used, but it really means multiply. If you want the moment generated by a 60 lb. child 5 ft out at the end of a seesaw, then you would calculate moment by multiplying the two together:

60*(1 lb) * 5*(1 ft.)

which would give you

60*5 * lb *ft

or

300 ft-lb

But it really should be thought of as

300*ft*lb


Also, if you look at units as things you multiply and divide, then it's easy to see how to convert units using identities. If you want to convert 10 feet per second to miles per hour, you start with:

10 * ft/sec

I can multiply this by 1 and not change it (1*a = a). I also know that

3600 * sec = 1 * hr

I can rewrite this as:

(3600 *sec) / (1*hr) = 1

Or, I could invert it to get:

(1*hr) /(3600 *sec) = 1


I can then use this as my "1" to multiply the 10 ft/sec. So:

10 ft./sec * (3600 sec)/(1 hr)

To get:

10*3600 * (ft * sec) / (sec*hr)

Since the units are all multiplied and divided, I can "cancel out" the sec/sec because anything divided by itself is just 1, and then get:

36000 ft/hr

This kind of formality might seem unnecessarily difficult because we are only half way there, but once you practice it a bit, it becomes easy. It is also very necessary when the problems and units get much more complex.

For example, how about the mass unit called a slug?

1 slug = 1 lbf*sec^2/ft

Where lbf is pounds(force) or a weight. If you don't know how to formally deal with units, you will be lost.