Application and Limitations of the Method of Dimensional Analysis

Dimensional analysis, built on the principle of homogeneity (1.8.2), has three practical applications -- and five genuine limits on what it can do.

1. Converting a quantity from one system of units to another. Since the product of a quantity's numerical value $n$ and its unit $[u]$ is a fixed physical quantity, $n_1[u_1]=n_2[u_2]$. Writing a quantity's dimensions as $a$ in mass, $b$ in length, $c$ in time, and the two systems' base units as $(M_1,L_1,T_1)$ and $(M_2,L_2,T_2)$: $n_1[M_1^aL_1^bT_1^c]=n_2[M_2^aL_2^bT_2^c]$, so

   $n_2=n_1\left(\frac{M_1}{M_2}\right)^a\left(\frac{L_1}{L_2}\right)^b\left(\frac{T_1}{T_2}\right)^c.$

   Worked example: convert $G_{\text{SI}}=6.6\times10^{-11}\ \text{N m}^2\ \text{kg}^{-2}$ to CGS units. $G$'s dimensional formula is $[M^{-1}L^3T^{-2}]$, so $a=-1,b=3,c=-2$. With $M_1=1$ kg, $L_1=1$ m, $T_1=1$ s (SI) and $M_2=1$ g $=10^{-3}$ kg, $L_2=1$ cm $=10^{-2}$ m, $T_2=1$ s (CGS): $G_{\text{cgs}}=6.6\times10^{-11}\times(10^3)^{-1}\times(10^2)^{3}\times1^{-2}=6.6\times10^{-11}\times10^{-3}\times10^{6}=6.6\times10^{-8}\ \text{dyne cm}^2\ \text{g}^{-2}$.
2. Checking the dimensional correctness of an equation. If every additive term reduces to the same dimensions, the equation is at least dimensionally consistent (though this does not guarantee it is numerically correct -- see limitation 5 below). Worked example: check $v=u+at$: $[LT^{-1}]\stackrel{?}{=}[LT^{-1}]+[LT^{-2}][T]=[LT^{-1}]+[LT^{-1}]$ -- both sides reduce to $[LT^{-1}]$, so the equation passes. Second example: check $\tfrac12mv^2=mgh$: LHS $=[M][LT^{-1}]^2=[ML^2T^{-2}]$; RHS $=[M][LT^{-2}][L]=[ML^2T^{-2}]$ -- both sides match, so the equation is dimensionally correct.
3. Establishing a relationship among physical quantities. If a quantity $Q$ is believed to depend on quantities $Q_1,Q_2,Q_3$ as $Q=k\,Q_1^aQ_2^bQ_3^c$ (with $k$ an unknown dimensionless constant), substituting each quantity's dimensional formula and matching powers of $M$, $L$, $T$ on both sides (using the principle of homogeneity) pins down $a$, $b$, $c$ -- though not $k$ itself. Worked example (simple pendulum): if the period $T$ depends on bob mass $m$, string length $l$, and $g$, write $T=k\,m^al^bg^c$. Dimensionally, $[T]=[M]^a[L]^b[LT^{-2}]^c=[M^aL^{b+c}T^{-2c}]$; matching powers gives $a=0$ (no $M$ on the LHS), $-2c=1\Rightarrow c=-\tfrac12$, and $b+c=0\Rightarrow b=\tfrac12$. So $T=k\,l^{1/2}g^{-1/2}=k\sqrt{l/g}$ -- and experimentally $k=2\pi$, recovering the familiar $T=2\pi\sqrt{l/g}$. Worked example (circular-motion force): if a force $F$ on a body moving in a circle of radius $r$ depends on mass $m$, speed $v$, and $r$ as $F=k\,m^av^br^c$ (with $k=1$): $[MLT^{-2}]=[M]^a[LT^{-1}]^b[L]^c=[M^aL^{b+c}T^{-b}]$; matching gives $a=1$, $-b=-2\Rightarrow b=2$, $b+c=1\Rightarrow c=-1$, so $F=mv^2r^{-1}=\dfrac{mv^2}{r}$ -- the familiar centripetal-force formula. Five limitations of dimensional analysis:

1. It gives no information at all about dimensionless constants in a formula (like $1,2,\ldots,\pi,e$) -- these have to come from elsewhere (experiment or a full derivation).
2. It cannot tell whether a given quantity is a vector or a scalar.
3. It is not suitable for deriving relations that involve trigonometric, exponential or logarithmic functions, since these are dimensionless functions whose argument must be dimensionless -- the method has nothing to say about their internal structure. …