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If either of these rules is violated, an equation is not dimensionally consistent and cannot possibly be a correct statement of physical law. This simple fact can be used to check for typos or algebra mistakes, to help remember the various laws of physics, and even to suggest the form that new laws of physics might take. This last use of dimensions is beyond the scope of this text, but is something you will undoubtedly learn later in your academic career.

Using dimensions to remember an equation

Suppose we need the formula for the area of a circle for some computation. Like many people who learned geometry too long ago to recall with any certainty, two expressions may pop into our mind when we think of circles: π r 2 and 2 π r . One expression is the circumference of a circle of radius r and the other is its area. But which is which?

Strategy

One natural strategy is to look it up, but this could take time to find information from a reputable source. Besides, even if we think the source is reputable, we shouldn’t trust everything we read. It is nice to have a way to double-check just by thinking about it. Also, we might be in a situation in which we cannot look things up (such as during a test). Thus, the strategy is to find the dimensions of both expressions by making use of the fact that dimensions follow the rules of algebra. If either expression does not have the same dimensions as area, then it cannot possibly be the correct equation for the area of a circle.

Solution

We know the dimension of area is L 2 . Now, the dimension of the expression π r 2 is

[ π r 2 ] = [ π ] · [ r ] 2 = 1 · L 2 = L 2 ,

since the constant π is a pure number and the radius r is a length. Therefore, π r 2 has the dimension of area. Similarly, the dimension of the expression 2 π r is

[ 2 π r ] = [ 2 ] · [ π ] · [ r ] = 1 · 1 · L = L,

since the constants 2 and π are both dimensionless and the radius r is a length. We see that 2 π r has the dimension of length, which means it cannot possibly be an area.

We rule out 2 π r because it is not dimensionally consistent with being an area. We see that π r 2 is dimensionally consistent with being an area, so if we have to choose between these two expressions, π r 2 is the one to choose.

Significance

This may seem like kind of a silly example, but the ideas are very general. As long as we know the dimensions of the individual physical quantities that appear in an equation, we can check to see whether the equation is dimensionally consistent. On the other hand, knowing that true equations are dimensionally consistent, we can match expressions from our imperfect memories to the quantities for which they might be expressions. Doing this will not help us remember dimensionless factors that appear in the equations (for example, if you had accidentally conflated the two expressions from the example into 2 π r 2 , then dimensional analysis is no help), but it does help us remember the correct basic form of equations.

Check Your Understanding Suppose we want the formula for the volume of a sphere. The two expressions commonly mentioned in elementary discussions of spheres are 4 π r 2 and 4 π r 3 / 3 . One is the volume of a sphere of radius r and the other is its surface area. Which one is the volume?

4 π r 3 / 3

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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