Deducing Relations Among Physical Quantities

Deducing Relations Among Physical Quantities

The method of dimensions can sometimes be used to derive the form of a relationship between physical quantities, provided you know which quantities the dependent variable depends on. The method assumes a product-type dependence — that is, the dependent quantity can be expressed as a product of powers of the independent quantities, multiplied by a dimensionless constant.

To find the unknown exponents, suppose a physical quantity $Q$ depends on three independent quantities $A$, $B$, and $C$. We write

where $k$ is a dimensionless constant and $a$, $b$, $c$ are exponents to be determined. Writing out the dimensions of each quantity on both sides and equating the powers of every base dimension present (mass, length, time, …) gives a set of simultaneous equations for $a$, $b$, and $c$. Solving these equations pins down the exponents, and hence the form of the relationship — though never the value of $k$ itself.

As an illustration, consider the time period $T$ of a simple pendulum, which we expect to depend on its length $l$, the mass of the bob $m$, and the acceleration due to gravity $g$. Writing $T = k\, l^{x} g^{y} m^{z}$ and comparing dimensions:

Equating powers of L, T, and M on both sides gives $x + y = 0$, $-2y = 1$, and $z = 0$, so $x = \tfrac{1}{2}$, $y = -\tfrac{1}{2}$, $z = 0$. This gives

with no dependence on the mass of the bob at all. The method of dimensions cannot tell us the value of $k$ — that has to come from a full derivation or from experiment; it turns out $k = 2\pi$.