1.3 Dimensions and Dimensional Analysis
Every physical quantity can be written as powers of mass, length and time. Both sides of a true equation must have the same dimensions — a rule that catches mistakes and even predicts how a quantity depends on others.
A student derives a formula for the period of a pendulum and gets $T=2\pi\sqrt{g/l}$. Is it right? You do not need to repeat the derivation to find out. The left side is a time, measured in seconds; the right side has units $\sqrt{(\text{m/s}^2)/\text{m}}=\text{s}^{-1}$. A time cannot equal "per second", so the formula is wrong — and a quick glance shows the fraction is upside down.
This kind of check is called dimensional analysis. It rests on one idea: every quantity in mechanics is built from mass, length and time, and an equation can only be true if both sides are built in the same way. Physicists use it constantly — to check their algebra, to convert units between systems, and, most strikingly, to guess how a quantity must depend on the others before doing any detailed theory.
Dimensions of a physical quantity
In Section 1.1 every derived unit was a product of powers of base units: $\text{N}=\text{kg}^1\,\text{m}^1\,\text{s}^{-2}$. If we forget which particular unit is used and keep only the kind of base quantity and its power, we get the dimensions of the quantity. We write $[\text{M}]$, $[\text{L}]$, $[\text{T}]$ for the dimensions of mass, length and time (and $[\text{A}]$, $[\text{K}]$, $[\text{mol}]$, $[\text{cd}]$ for the other base quantities when needed).
The dimensions of a physical quantity are the powers to which the base quantities must be raised to represent it. The expression showing them is its dimensional formula; for force, $[\text{M}\,\text{L}\,\text{T}^{-2}]$. An equation stating it, such as $[F]=[\text{M}\,\text{L}\,\text{T}^{-2}]$, is a dimensional equation. Square brackets $[Q]$ mean "the dimensions of $Q$".
A quantity whose dimensions are $[\text{M}^0\text{L}^0\text{T}^0]$ is dimensionless: a pure number, a ratio of like quantities (strain, refractive index, relative density), an angle in radians, or the argument of $\sin$, $\cos$, $\log$ or $e^x$.
To find a dimensional formula, start from any equation that defines the quantity and replace every quantity by its dimensions. Dimensions multiply and divide exactly like algebra; pure numbers ($\tfrac12$, $2\pi$) are dimensionless and drop out.
| Quantity | Defining relation | Dimensional formula |
|---|---|---|
| velocity, speed | distance ÷ time | $[\text{L}\,\text{T}^{-1}]$ |
| acceleration | velocity ÷ time | $[\text{L}\,\text{T}^{-2}]$ |
| force | $F=ma$ | $[\text{M}\,\text{L}\,\text{T}^{-2}]$ |
| momentum, impulse | $p=mv$, $J=F\,\Delta t$ | $[\text{M}\,\text{L}\,\text{T}^{-1}]$ |
| work, energy, torque | $W=Fd$, $\tau=rF$ | $[\text{M}\,\text{L}^2\text{T}^{-2}]$ |
| power | $P=W/t$ | $[\text{M}\,\text{L}^2\text{T}^{-3}]$ |
| pressure, stress | $p=F/A$ | $[\text{M}\,\text{L}^{-1}\text{T}^{-2}]$ |
| density | $\rho=m/V$ | $[\text{M}\,\text{L}^{-3}]$ |
| frequency, angular speed | $\nu=1/T$, $\omega=\theta/t$ | $[\text{T}^{-1}]$ |
| spring constant, surface tension | $k=F/x$, $S=F/l$ | $[\text{M}\,\text{T}^{-2}]$ |
| gravitational constant | $F=\tfrac{Gm_1m_2}{r^2}$ | $[\text{M}^{-1}\text{L}^3\text{T}^{-2}]$ |
| Planck constant | $E=h\nu$ | $[\text{M}\,\text{L}^2\text{T}^{-1}]$ |
| strain, angle | length ÷ length, arc ÷ radius | $[\text{M}^0\text{L}^0\text{T}^0]$ |
Some quantities with dimensions are constants of nature, such as $G$, $h$ and the speed of light $c$: they are dimensional constants. Numbers such as $2\pi$ are dimensionless constants.
Checking equations: the principle of homogeneity
We can only add, subtract or equate quantities of the same kind: $3\ \text{kg}+2\ \text{m}$ means nothing. This is the foundation of every dimensional check.
The converse is false: a dimensionally consistent equation need not be correct, because dimensionless factors ($\tfrac12$, $2\pi$, $3$) are invisible to the check.
Deducing relations between quantities
Suppose we believe that a quantity $Q$ depends only on a few other quantities, but we do not know how. If we assume a power law $Q=k\,x^a y^b z^c$ with a dimensionless constant $k$, the principle of homogeneity gives one equation for each base dimension. With mass, length and time we get three equations — enough to find three unknown powers.
- List the quantities $x$, $y$, $z$ on which $Q$ may depend (at most three if you use only M, L, T).
- Write $Q=k\,x^a y^b z^c$ with $k$ dimensionless.
- Replace each quantity by its dimensional formula and collect the powers of $\text{M}$, $\text{L}$ and $\text{T}$ on the right.
- Equate the powers on both sides: three linear equations in $a$, $b$, $c$. Solve them.
- Write the result. The constant $k$ must come from experiment or a full theory.
Converting units between systems
Dimensions also convert a numerical value from one unit system to another. If a quantity has dimensions $[\text{M}^a\text{L}^b\text{T}^c]$, then $n_1u_1=n_2u_2$ with $u=M^aL^bT^c$ gives $$n_2=n_1\left[\frac{M_1}{M_2}\right]^a\left[\frac{L_1}{L_2}\right]^b\left[\frac{T_1}{T_2}\right]^c ,$$ where $M_1$, $L_1$, $T_1$ are the units of mass, length and time of the first system and $M_2$, $L_2$, $T_2$ those of the second.
1. No dimensionless constants. It gives $T\propto\sqrt{l/g}$ but never the $2\pi$; it cannot even tell whether a dimensionless variable such as the amplitude angle enters (it does, for large swings — see Interactive 2).
2. Only power laws. It cannot produce $\sin$, $\exp$ or $\log$ dependences, nor equations with several terms added together, such as $x=ut+\tfrac12at^2$.
3. At most three unknowns with M, L, T alone. If $Q$ depends on four or more quantities, the equations do not determine all the powers.
4. Same dimensions, different quantities. Work and torque are both $[\text{M}\,\text{L}^2\text{T}^{-2}]$; a correct dimensional check cannot tell which one you have.
5. You must guess the right variables. Leave out one that matters (or include one that does not) and the method gives a confident wrong answer.
Exercises
- 1 momentum
- 2 pressure
- 3 power
- 4 impulse $F\,\Delta t$ (compare with part 1)
- 5 the spring constant $k$ in $F=kx$
- 6 kinetic energy $\tfrac12mv^2$ and work $Fd$ (compare them)
- 7 angular speed $\omega=\theta/t$
- 8 the Planck constant $h$, from $E=h\nu$
- 1 $v=u+at$
- 2 $s=ut+at$
- 3 $K=\tfrac12mv^2$
- 4 $T=2\pi\sqrt{g/l}$
- 5 $p=\rho gh$
- 6 $v=\sqrt{\dfrac{Fl}{m}}$ for a wave on a string of tension $F$, length $l$ and mass $m$
- 7 Show that $s=ut+at^2$ is dimensionally consistent, and explain why it is nevertheless wrong.
- 1 The time $t$ for an object to fall from rest through a height $h$ depends on $h$, $g$ and its mass $m$.
- 2 The frequency $\nu$ of a stretched string depends on its length $l$, tension $F$ and mass per unit length $\mu$ ($[\mu]=[\text{M}\,\text{L}^{-1}]$).
- 3 The speed of sound $v$ in a gas depends on its pressure $p$ and density $\rho$.
- 4 The energy $E$ associated with a mass $m$ depends on $m$ and the speed of light $c$.
- 5 The period $T$ of a mass $m$ on a spring of spring constant $k_s$.
- 6 The period $T$ of a satellite in a circular orbit depends on the orbit radius $r$, the planet's mass $M$ and $G$. What law of planetary motion do you obtain?
- 1 How many dynes ($1\ \text{dyne}=1\ \text{g cm s}^{-2}$) make one newton?
- 2 Convert a density of $1\ \text{kg/m}^3$ into $\text{g/cm}^3$.
- 3 $G=6.67\times10^{-11}\ \text{N m}^2\text{kg}^{-2}$. Find its value in CGS units ($\text{cm}^3\text{g}^{-1}\text{s}^{-2}$).
- 4 In a new system the unit of mass is $100\ \text{kg}$, of length $10\ \text{m}$ and of time $1\ \text{min}$. What is $1\ \text{J}$ in the new unit of energy?
- 5 Why can dimensional analysis not derive $y=A\sin(\omega t)$?
- 6 Torque and work have the same dimensions. Are they the same physical quantity? Explain.
E1. Board · 3 marks State the principle of homogeneity of dimensions. Use it to check $x=ut+\tfrac12at^2$. Give one limitation of the method.
E2. NEET The dimensional formula of energy per unit volume is (a) $[\text{M}\,\text{L}\,\text{T}^{-2}]$ (b) $[\text{M}\,\text{L}^{-1}\text{T}^{-2}]$ (c) $[\text{M}\,\text{L}^2\text{T}^{-2}]$ (d) $[\text{M}\,\text{L}^{-2}\text{T}^{-2}]$
E3. JEE Main A force varies as $F=a\sqrt x+bt^2$, where $x$ is a distance and $t$ a time. The dimensions of $\dfrac ab$ are (a) $[\text{L}^{1/2}\text{T}^{-2}]$ (b) $[\text{L}^{-1/2}\text{T}^{2}]$ (c) $[\text{L}\,\text{T}^{-2}]$ (d) $[\text{L}^{-1/2}\text{T}^{-2}]$
E4. JEE Main NEET The speed $v$ of waves on deep water may depend on the wavelength $\lambda$, the acceleration due to gravity $g$ and the density $\rho$ of water. Dimensional analysis gives $v$ proportional to (a) $\sqrt{g\lambda}$ (b) $\sqrt{g/\lambda}$ (c) $g\lambda$ (d) $\sqrt{\lambda/g}$
E5. JEE Advanced (One or more correct) Using the speed of light $c$, the gravitational constant $G$ and the Planck constant $h$ as fundamental constants, which of the following have the stated dimensions? (A) $\sqrt{hG/c^3}$ is a length (B) $\sqrt{hG/c^5}$ is a time (C) $\sqrt{hG/c}$ is a mass (D) $\sqrt{hc^5/G}$ is an energy
Research question. How fast can you walk before you have to run — on the Earth, on the Moon, or as a dinosaur?
Why it matters. Biomechanics researchers compare animals of very different sizes using the dimensionless Froude number $Fr=\dfrac{v^2}{gL}$, where $L$ is leg length. Humans, horses and birds all switch from walking to running at roughly $Fr\approx0.5$. Palaeontologists use the same idea to estimate dinosaur speeds from fossil footprints, and space agencies used it to understand why Apollo astronauts hopped instead of walking.
What to do.
- Show that $Fr$ is dimensionless. Explain, using this section, why the walk–run transition speed must be of the form $v=k\sqrt{gL}$.
- Taking $Fr=0.5$, find the transition speed for an adult ($L=0.90\ \text{m}$) and a child ($L=0.50\ \text{m}$) on the Earth.
- Repeat for the adult on the Moon ($g=1.62\ \text{m/s}^2$). Compare with video of astronauts moving on the Moon.
- Measure your own leg length and the speed at which you naturally break into a run (time yourself over $20\ \text{m}$, speeding up gradually). Compute your $Fr$.
Tools: Tape measure, stopwatch or phone timer, a corridor or field.
Hand in: A table of predicted and measured transition speeds and a paragraph on what dimensional analysis can and cannot tell you here.
Going further. Look up how stride length divided by leg length is used to estimate the speed of a dinosaur from its trackway. What extra assumption, beyond dimensions, is needed?
🧪 Self-check quiz
Q1. The dimensional formula of force is
Q2. Which of these quantities is dimensionless?
Q3. Which pair has the same dimensions?
Q4. For a pendulum, $T=2\pi\sqrt{l/g}$. Which part of this result can dimensional analysis not give?
Q5. One joule is equal to