Equations

Unit Conversion: Dimensional analysis

by Donald G Wiggins, M.A./M.S.

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.

On September 23, 1999, NASA's $125 million Mars Climate Orbiter approached the red planet under guidance from a team of flight controllers at the Jet Propulsion Laboratory. The probe was one of several planned for Mars exploration, and would stay in orbit around the planet as the first extraterrestrial weather satellite. It had been in flight for over nine months, covering more than 415 million miles of empty space on its way to Mars. As the Orbiter reached its final destination, the flight controllers began to realize that something was wrong. They had planned for the probe to reach an orbit approximately 180 km off the surface of Mars – well beyond the planet's thin atmosphere. But new calculations based on the current flight trajectory showed the Orbiter skimming within 60 km of the Martian surface. Now the probe would actually enter the planet's thin atmosphere, something for which it was never designed. The consequences were catastrophic: when the scientists and engineers commanding the probe lost communication, they could only assume that the spacecraft was incinerated by the friction from an atmospheric entry that it was never supposed to make.

Figure 1: An artists rendition of the Mars Climate Orbiter.

What caused this disaster? The problem arose in part from a simple, seemingly innocent, mistake. Throughout the journey from Earth, solar winds pushed against the solar panels of the probe, throwing the spacecraft off course by a small amount. The designers had planned for this, and jet thrusters were turned on by the flight controllers to apply a force, making numerous small corrections to readjust its course. Unfortunately, the NASA engineers measured this force in pounds (a non-metric unit), while the JPL team worked in Newtons (a metric unit), and the software that calculated how long the thrusters should be fired did not make the proper conversion. Since 1 pound = 4.45 Newtons, 4.45 times too much thrust was applied each time the thrusters were used. While each individual adjustment mistake was very small, this mistake grew larger and larger over multiple adjustments, resulting in the craft's premature demise in the Martian atmosphere.

The Orbiter loss illustrates the need for consistent use of units. Most people, however, are most comfortable working in whatever units they grew up using. As a result, unit consistency may not be possible within or between teams around the world. Ideally, people should be comfortable with a variety of ways of converting units in order to allow for collaboration among individuals from a variety of backgrounds.

While most people are not controlling NASA space probes, unit conversion is something that happens every day, in all walks of life. Even such a simple problem as figuring out that two dozen eggs equals 24 eggs is, at its heart, a unit conversion problem. Whether you realize it or not, when you do this problem in your head, you're figuring it out like this:

$$2 \, \cancel{dozen} eggs \times \frac{\text{12 eggs}}{1 \cancel{dozen} eggs} = \text{ 24 eggs}$$
Comprehension Checkpoint
Due to the lack of a common unit for calculations, the Mars Climate Orbiter
Incorrect.
Correct!

Dimensional Analysis

Generally, unit conversions are most easily solved using a process called dimensional analysis, also known as the factor-label method. A notable exception is the conversions among temperature units (see our Temperature module for details). Dimensional analysis uses three fundamental facts to make these conversions, which lead to the steps in the conversion process:

1. A conversion factor is a statement of the equal relationship between two units. The first step in dimensional analysis is therefore identifying the conversion factor(s) you will need to make your conversion. In the egg problem, the statement that "1 dozen eggs = 12 eggs" is a conversion factor.

2. If you multiply by a conversion factor in the form of a ratio, you are really only multiplying by 1, since the two parts of the ratio equal each other:

$$\frac{12 \, eggs}{1 \, \text{dozen eggs}} or \frac{1 \, \text{dozen eggs}}{12 \, eggs}$$

The second step in dimensional analysis is therefore to set up a mathematical problem that uses one or more conversion factors to get to the units you are interested in. In the egg problem, if you have 2 dozen eggs and want to know how many individual eggs you have, you would set up the problem like this:

$$2 \, \text{dozen eggs} \times \frac{12 \, eggs}{1 \, dozen \, eggs}$$

3. Units, just like numbers or variables, "cancel" when you divide a unit by itself. So the final step in dimensional analysis is to work the math problem you've set up, canceling units along the way. In the egg example, the "dozen eggs" in the bottom of the ratio cancels the "dozen eggs" in your original number, leaving "eggs" as the only unit left in the problem, as shown in the final answer, 24 eggs.

$$2 \, \cancel{dozen} \text{eggs} \times \frac{12 \, eggs}{1 \, \cancel{dozen} \, eggs} = 24 \, eggs$$

Let's apply these steps to a slightly more complex problem than counting eggs? How much money would it cost to fill a truck's 23 gallon gas tank if gas cost $2.87 per gallon?

First, create the conversion factor. Given the price, you can say 1 gallon = $2.87. Then go through and set-up the equations:

$$23 \, gal \times \frac{$2.87}{1 \, gal} =$$
$$23 \, \cancel{gal} \times \frac{$2.87}{1 \, \cancel{gal}} = $66.01$$

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.

$$\frac{100 \cancel{km}}{1 \, hr} \times \frac{1 \, mile}{1.61 \cancel{km}} = \frac{62.1 \, mi}{1 \, hr} = 62.1 \, mph$$
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:

$$\frac{6.50 \, \cancel{pesos}}{1 \, \cancel{L}} \times \frac{8.95 \, \cancel{c}}{1 \, \cancel{peso}} \times \frac{$1}{100 \cancel{c}} \times \frac{3.79 \, \cancel{L}}{1 \, gal} = \frac{$2.20}{1 \, gal} = $2.20 \, \text{per gallon}$$

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.

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