Uses of Dimensional Analysis

Dimensional analysis — comparing the dimensional formulae of the different terms in a physical relationship — turns out to be useful in three distinct ways:

1. Checking whether a physical equation is correct. In any valid equation relating physical quantities, the dimensions of every term on both sides must be identical; this requirement is called the principle of homogeneity of dimensions. For example, consider the first equation of motion, $v = u + at$. The dimensions of each term are: $[v] = [LT^{-1}]$, $[u] = [LT^{-1}]$, and $[at] = [LT^{-2}][T] = [LT^{-1}]$. Since the L.H.S. has dimensions $[LT^{-1}]$ and every term on the R.H.S. also has dimensions $[LT^{-1}]$, the equation is dimensionally homogeneous, and therefore dimensionally correct (though homogeneity alone does not guarantee the equation is physically correct — it only rules out certain kinds of errors).
2. Deriving the form of a relationship between physical quantities. As an example, consider the period $T$ of oscillation of a simple pendulum, which we expect to depend on its length $l$ and the acceleration due to gravity $g$. Suppose $T \propto l^a$ and $T \propto g^b$, so that $T \propto l^a g^b$, i.e. $T = k\, l^a g^b$, where $k$ is a dimensionless constant of proportionality and $a, b$ are numbers to be found. Equating dimensions on both sides: $[L^0M^0T^1] = k\,[L^1]^a[LT^{-2}]^b = k\,[L^{a+b}T^{-2b}]$. Comparing powers of $L$ and $T$ on both sides gives $a + b = 0$ and $-2b = 1$, so $b = -1/2$ and $a = -b = 1/2$. Hence $T = k\,l^{1/2}g^{-1/2} = k\sqrt{l/g}$. Dimensional analysis alone cannot determine the dimensionless constant $k$ — that has to come from experiment (or from solving the pendulum's equation of motion exactly), and it is found to be $k = 2\pi$, giving the familiar formula $T = 2\pi\sqrt{l/g}$. …