Physical dimensions, such as length, mass and time, are measured through a variety of units. Take mass as an example; it can be measured in kilograms, grams, moles, pounds, ounces, tons and so on. It is convenient to find a simple and reliable way to convert between each of these different units. A conversion factor is just a fraction equal to one that relates two units together. For example, we know that one inch (in) measures the same length as 2.54 centimeters; therefore, we can construct conversion factors from this fact as follows
As you can see, two fractions can be constructed from the starting equation. The important thing to keep in mind is that both fractions simplify to one; hence, multiplying an expression by either of these fractions will not change the expression. The first fraction allows us to change from centimeters to inches while the second factor changes inches to centimeters. One does not need to memorize which conversion factor needs to be used in a particular problem; to figure out which one to use, we can always invoke dimensional analysis.
Very often, multiple conversion factors are needed in succession to achieve the desired units. In this situation, the knowledge of dimensional analysis coupled with conversion factors becomes extremely useful at minimizing potential conversion mistakes. Let us look at an example to illustrate this point.
A car is traveling at 75 mph (miles per hour). Find this speed in SI units (m/s). 1 mile is 1.61 km.
In this example, we used three conversion factors. The equation holds because each time we use a conversion factor we are multiplying by one. Furthermore, notice that after simplifying, the only unit that survives in the numerator is the meter and the only unit that survives in the denominator is the second (this is what we wanted). Had we picked the wrong conversion factor, then we would have ended up with different units. Imagine we did the following instead;
This is mathematically correct (remember we just multiplied by one three times);however the units look very awkward. We don’t say a car is traveling at a speed of 12.94 meters mile squared per second kilometer squared, instead we say that the car is traveling at 75 mile per hour or 36 meters per second. This is why when using multiple conversion factors, it is always a good idea to keep track of all the units (dimensional analysis) to make sure that the simplification yields what you are looking for.