Changing units

Assigning units

When we are creating a physical equation, typically we are writing equations that say two physical quantities are equal. Thus, we can say the length of two objects is the same. We would then write an equation

                   $$ L_1 = L_2$$

Note that this equation will be true no matter what unit we choose — as long as we have chosen the same standard — the same unit —for each. But if we tried to equate a length to a time, say, to consider 1 meter = 1 second, then we would be in trouble. The two measurements are made with different kinds of measuring devices and with different standards. It doesn't make sense to say that a distance equals to a time because if we changed one of our units, the equation would no longer hold.

The crucial results are

We can only equate quantities that have the same dimensionality.

Even if you have made sense of measurement — how we assign numbers to physical quantities, and to dimensionality — how we describe the different kinds of measurements we make, you still have to figure out how to attach units to your measurements in a consistent way.

This sounds kind of trivial. "If I measure something in centimeters and find 100 of them, I just write 100 cm.  If I measure it in meters, I'll just get 1 and will write 1 m. What's the big deal?"

It might seem trivial, but there are subtleties. We know that we can represent the length of a meter stick by reporting its length either as 100 cm or as 1 m.  So - since 100 cm and 1 m have the same dimension, and represent the same physical system (a meter stick), I can equate them

100 cm = 1 m

This is a bit strange if you haven't learned to pay attention to the units. It looks like you are saying "100 = 1" but you're not. You're saying "100 of these things (called cm) is equal to 1 of these other things (called m)." NOT the same as "100 = 1". You have to keep in mind that you're putting an equal sign between "things" (in this case, lengths) not simply numbers.

We can only expect the numbers in front of the unit of both sides to be the same if we use the same units.

Changing units

We will often be writing equations in which we'll be given values in mixed units. A density might be given in grams/cm³, and we might wind up applying it to the size of a cell given in micrometers (μm). Even if our dimensions match, how can we convert so that our units are consistent?

Here's the trick. An equation says that the two sides are the same, so I know that if we divide both sides by the same thing I should still get the same thing. That still seems OK. What if I take my (legitimate) definition of the centimeter above and divide each side by "1 m"? Then I get

(100 cm) / (1 m) = 1.

This turns out to be immensely useful. I can multiply anything by 1 and not change it. So can I just put in factors of 100 anywhere? The answer is, "Yes, I can — as long as I also keep the cm/m part." That might make things more complicated, but it also might make things easier.

For example, suppose you're given a problem that has mixed units. Here's the key principle.

To change units, multiply by 1 in the appropriate form — taking the ratio of the same thing expressed in different ways.

As an example, to change 60 miles/hour to meters/second we can write the chain:

Note what we've done. In the first line, each set of parentheses represents "1" — the top and the bottom are the same physical quantity represented in different ways. And we've chosen to write them so that the unit in the top is cancelled by the one at the bottom until we get to the quantity we want. We then put all the numbers together and all the units together. We multiply the numbers on our calculator to get a final number and we cross out all the top/bottom places where the units are the same to get a final unit.*

Expressing things in different units is valuable because it helps us think about the same thing in different ways. For the example above, thinking that your car is going 60 miles/hour is useful if you're thinking about how long it's going to take you to drive 120 miles on the highway — or whether you might get a speeding ticket in a 35 mile/hour zone! But 27 m/s is important if you're thinking about how far you travel in one second — if you're figuring out how far you might travel before you start to brake (in order to figure out when you're too close to another car on the highway).

* Note the grammatical peculiarity: Because I've written the unit out instead of just using the symbol, I have to write "5280 feet" but "1 foot".  The unit is the same and we can cancel them despite the fact that they look different when written out. That's a good reason for using an abbreviation — "s" for either "second" or "seconds". They mean the same thing here.

Joe Redish and Wolfgang Losert 8/29/12