Equations

Unit Conversion: Dimensional analysis

Did you know that failure of scientists to use the same unit of measurement resulted in the loss of a $125 million satellite that was supposed to give us weather reports from Mars? This disaster could have been avoided through better communication along with unit conversion by dimensional analysis. “Dimensional analysis” may sound complicated, but this is a method we use in everyday conversions, such as when figuring out how many gallons of gas we can get for$30 or how many donuts are in two dozen.

First, create the conversion factor. Given the price, you can say 1 gallon = $2.87. Then go through and set-up the equations: Now that you've filled your tank, it's time to head off for your day trip to Mexico. As you cross the border from the US into Mexico, you notice that the speed limit sign reads 100. Wow! Can you step on the gas, or is there something else going on here? There are very few countries other than the United States where you will find speeds in miles per hour – almost everywhere else they would be in kilometers per hour. So, some converting is in order to know what the speed limit would be in a unit you're more familiar with. First, we need to define what "100 kilometers per hour" means mathematically. The "per" tells you that the number is a ratio: 100 kilometers distance per 1 hour of time. Other than that, you need to know the conversion factor between kilometers and miles, namely 1 mile = 1.61 km. Now the set-up is pretty simple. Give it a try yourself, and then run the animation below (an SWF file, Flash required) or see the equation to reveal the math needed to solve the problem. Comprehension Checkpoint Dimensional analysis is a method that can be used to convert km/h to mph. Correct! Incorrect. Conversions of multiple units So far you've seen examples with only one conversion factor, but this method can be used for more complicated situations. When it's time to leave for home from your day trip in Mexico, you realize you have just enough gas to make it back across the border into the US before you have to fill up. You notice that you could buy gas for 6.50 pesos per liter before you head home. At first glance that seems more expensive than the$2.87 per gallon at home, but is it really? You need to convert to be sure. Fortunately you came prepared, and looked up the currency exchange rate (1 peso = 8.95 cents) and volume conversion (1 gallon = 3.79 Liters) the morning before you left.

This conversion is more complicated than the previous examples for two reasons. First, imagine that you do not have a single direct conversion factor for the monetary conversion (pesos to dollars). You know that 1 peso = 8.95 cents, and you also know that 100 cents = 1 dollar. Together, these two facts will let you convert the currency. The second twist is that you are not only changing the money unit – you also need to convert the volume unit as well. These two conversions can be done in a single set-up. The order does not matter, but both must be done. Try to set this one up for yourself first, and then run the animation (an SWF file, Flash required) or see the equation below to reveal the solution:

Notice that when set up properly, the "L" had to be placed above the division bar in the conversion factor in order to cancel out the "L" below the division bar in the original number. Also note that even though the "L" terms are separated by two conversion factors, they still cancel each other out. Now it is easier to decide whether you should fill up before or after you return to the United States.

You can see that you don't have to be an engineer at NASA to need dimensional analysis. You need to convert units in your everyday life (to budget for gas price increases, for example) as well as in scientific applications, like stoichiometry in chemistry and calculating past plate motions in geology. If you know what units you have to work with, and in what units you want your answer to be, you don't need to memorize a formula. If the teams working on the Mars Climate Orbiter had realized that they needed to go through these steps, we would be getting weather forecasts for Mars today.

Donald G Wiggins, M.A./M.S. “Unit Conversion” Visionlearning Vol. MAT-3 (2), 2008.