Conversions

Often in physics it is convenient to convert from one system of measurements to another or to convert within a system of measurements. This is done by using Conversion Factors. Converting units is typically very simple. The most important thing to remember is that you must apply the same number of conversion factors as there are dimensions. Its difficult to describe, but see the examples below for clarification. The best way to insure that the conversion you are performing is correct is to monitor your units very carefully. 

The Basics

Conversion factors usually come in the form x units_a = y units_b, where units_a and units_b are different units. Example: 1 yd = 3 ft. This can be read as "One yard equals three feet." The form of the conversion factor that is used to convert from one set of units to another set of units is found by replacing the equal sign by a division sign. Example: [ 1 yd / 3ft.] This read as a fraction; "One yard per three feet." and it is equivalent to its inverse, [ 3 ft / 1 yd ] or "three feet per yard." 

The Technical Proof

Remember how in elementary school you learned that you could multiply anything by one and it would still be itself? And that if you divided something by itself, the answer was one? This is the principle under which conversion factors operate. 

 3 ft = 1 yd

Now divide both sides by 1 yd and you get

[3 ft / 1yd] = [1 yd / 1yd]

[3 ft / 1 yd] = 1

If you try to divide both sides of the original equation by 3 ft, you will get

[3 ft / 3 ft] = [1 yd / 3 ft]

1 = [1 yd / 3 ft]

Because both of these equations are equal to one, you can use them to convert from one system of measurements to another, by multiplying in the conversion factor. This is true for ALL conversion factors: they are all equal to one, which is why they work.

Examples 1 & 2 - One-dimensional Conversions 

1. Convert 4 meters to yards. 

4 m = ? yd

According to the conversion factor chart, 1 m = 3.28 ft. So, we will convert from meters to feet and then we can convert from feet to yards, using the conversion factor 1 yd = 3 ft

(4 m) x ( 3.28 ft/ 1 m) = 13.12 ft = ? yd

Note that meters is put on the bottom of the conversion factor so that it will cancel when the multiplication is completed. Now to convert feet to yards.

(13.12 ft) x (1 yd / 3 ft) = 4.37 yd

So our final result is:

4 m = 4.37 yd

Also note that this can be combined into one step rather easily. 

4 m = ? yd

(4 m) x ( 3.28 ft/ 1 m) x (1 yd / 3 ft) = 4.37 yd

In one step, both the units of meters and the units of yards cancels rather nicely

2. Convert 30 centimeters to meters.

30 cm = ? m 

Examples 3 & 4 - Two-dimensional Conversions 

3. Convert 15 square meters to square kilometers.

4. Convert 3 square kilometers to square miles

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