Exercise 6

The advice about apples and oranges is not right. Consider the familiar equations for a circle $C=2\pi r$ and $A=\pi r^{2}$.

• Check that $C$ and $A$ have different dimensional formulas.

• Produce an equation that is not dimensionally homogeneous (i.e., it adds apples and oranges) but is nonetheless true of any circle.

• The prior item asks for an equation that is complete but not dimensionally homogeneous. Produce an equation that is dimensionally homogeneous but not complete.

(Just because the old saying isn’t strictly right doesn’t keep it from being a useful strategy. Dimensional homogeneity is often used to check the plausibility of equations used in models. For an argument that any complete equation can easily be made dimensionally homogeneous, see , Chapter I, especially page 15.)