Dimensional Analysis

Dimensions express how a physical quantity depends on the base quantities mass $[M]$, length $[L]$ and time $[T]$ (and sometimes current, temperature). Dimensional analysis lets you check the correctness of an equation, derive the form of a relation up to a constant, and convert quantities between unit systems without memorising conversion factors.

Concept

Every term in a valid physical equation must have the same dimensions (principle of homogeneity). For instance, in $v = u + at$, each side has dimension $[L][T]^{-1}$. The dimensional formula of a quantity $Q$ is written as $[M^{a} L^{b} T^{c}]$.

Common formulas to memorise:

• Velocity: $[L T^{-1}]$
• Acceleration: $[L T^{-2}]$
• Force: $[M L T^{-2}]$
• Energy/Work: $[M L^{2} T^{-2}]$
• Power: $[M L^{2} T^{-3}]$
• Pressure: $[M L^{-1} T^{-2}]$

Formula Derivation

Suppose the time period $T$ of a simple pendulum depends on length $\ell$, mass $m$ and gravity $g$:

$T = k \, \ell^{a} m^{b} g^{c}$

Equating dimensions on both sides ($[T] = [L]^{a}[M]^{b}[L T^{-2}]^{c}$):

$[M^{0} L^{0} T^{1}] = [M^{b} L^{a+c} T^{-2c}]$

Comparing powers: $b=0$, $a+c=0$, $-2c=1 \Rightarrow c=-1/2$, $a=1/2$.

$T = k \sqrt{\frac{\ell}{g}}$

Experiment fixes $k = 2\pi$, giving $T = 2\pi \sqrt{\ell/g}$.

NEET-style Worked Example

The viscous force on a sphere of radius $r$ moving with velocity $v$ in a fluid of viscosity $\eta$ is $F = k \eta^{a} r^{b} v^{c}$. Find $a, b, c$.

Dimensions: $[\eta] = [M L^{-1} T^{-1}]$, $[r] = [L]$, $[v] = [L T^{-1}]$, $[F] = [M L T^{-2}]$.

$[M L T^{-2}] = [M^{a} L^{-a+b+c} T^{-a-c}]$

So $a=1$, $-a-c=-2 \Rightarrow c=1$, $-a+b+c=1 \Rightarrow b=1$.

$F = k\,\eta r v$

This is Stokes' law (experimentally $k = 6\pi$).

Common Confusions

• Dimensional analysis cannot find dimensionless constants like $2\pi$ or $1/2$.
• It cannot distinguish terms that have the same dimensions but different physics (e.g. work and torque both are $[M L^{2} T^{-2}]$).
• Trigonometric, exponential and log arguments must always be dimensionless.

Key Takeaways

• Both sides of any physical equation must have identical dimensions.
• Use dimensional analysis to spot wrong formulas and derive proportionalities.
• Constants of integration and pure numbers are invisible to dimensions.