Dimensional Analysis: Know Your Units

Dimensional analysis is a method of using the known units in a problem to help deduce the process of arriving at a solution. These tips will help you apply dimensional analysis to a problem.

How Dimensional Analysis Can Help

In science, units such as meter, second, and degree Celsius represent quantified physical properties of space, time, and/or matter. The International System of Measurement (SI) units that we use in science consist of seven base units, from which all other units are derived.

This means that a good knowledge of the units you're using for a problem can help you figure out how to approach a science problem, especially early on when the equations are simple and the biggest hurdle is memorization. If you look at the units provided within the problem, you can figure out some ways that those units relate to each other and, in turn, this might give you a hint as to what you need to do to solve the problem. This process is known as dimensional analysis.

A Basic Example

Consider a basic problem that a student might get right after starting physics. You're given a distance and a time and you have to find the average velocity, but you're completely blanking on the equation you need to do it.

Don't panic.

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If you know your units, you can figure out what the problem should generally look like. Velocity is measured in SI units of m/s. This means that there is a length divided by a time. You have a length and you have a time, so you're good to go.

A Not-So-Basic Example

That was an incredibly simple example of a concept that students are introduced to very early in science, well before they actually begin a course in physics. Consider a bit later, however, when you've been introduced to all kinds of complex issues, such as Newton's Laws of Motion and Gravitation. You're still relatively new to physics, and the equations are still giving you some trouble.

You get a problem where you have to calculate the gravitational potential energy of an object. You can remember the equations for force, but the equation for potential energy is slipping away. You know it's kind of like force, but slightly different. What are you going to do?

Again, a knowledge of units can help. You remember that the equation for gravitational force on an object in Earth's gravity and the following terms and units:

F_g = G * m * m_E / r²

• F_g is the force of gravity - newtons (N) or kg * m / s²
• G is the gravitational constant and your teacher kindly provided you with the value of G, which is measured in N * m² / kg²
• m & m_E are mass of the object and Earth, respectively - kg
• r is the distance between the center of gravity of the objects - m 
• We want to know U, the potential energy, and we know that energy is measured in Joules (J) or newtons * meter 
• We also remember that the potential energy equation looks a lot like the force equation, using the same variables in a slightly different way

In this case, we actually know a lot more than we need to figure it out. We want the energy, U, which is in J or N * m. The entire force equation is in units of newtons, so to get it in terms of N * m you will need to multiply the entire equation a length measurement. Well, only one length measurement is involved - r - so that's easy. And multiplying the equation by r would just negate an r from the denominator, so the formula we end up with would be:

F_g = G * m * m_E / r

We know the units we get will be in terms of N*m, or Joules. And, fortunately, we did study, so it jogs our memory and we bang ourselves on the head and say, "Duh," because we should have remembered that.

But we didn't. It happens. Fortunately, because we had a good grasp on the units we were able to figure out the relationship between them to get to the formula that we needed.

A Tool, Not a Solution

As part of your pre-test studying, you should include a bit of time to make sure you're familiar with the units relevant to the section you're working on, especially those that were introduced in that section. It is one other tool to help provide physical intuition about how the concepts you're studying are related. This added level of intuition can be helpful, but it shouldn't be a replacement for studying the rest of the material. Obviously, learning the difference between gravitational force and gravitational energy equations is far better than having to re-derive it haphazardly in the middle of a test.

The gravity example was chosen because the force and potential energy equations are so closely related, but that isn't always the case and just multiplying numbers to get the right units, without understanding the underlying equations and relationships, will lead to more errors than solutions.