1.4 Dimensional analysis  (Page 2/6)

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

Strategy

Solution

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

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

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

Significance