Foundations — Dimensional analysis — checking equations, deriving relations

This page assumes nothing. Before you can check equations or derive relations on the parent topic, you need a small toolkit of symbols and pictures. We build each one from the ground up, and each new idea leans only on the ones before it.

0. What is a "physical quantity"?

We need this because the entire topic is about keeping those labels straight.

1. The three base kinds: $M$, $L$, $T$

The square brackets are shorthand. When you see $[M]$, read it aloud as "the kind mass." These are the raw colours out of which every other colour is mixed.

Look at the three coloured bars in the figure: they are the primary paints. A blue bar for length, a yellow bar for time, a red bar for mass. Every other physical quantity in mechanics is some mixture of these three, in some amounts — and those amounts are exactly what a dimension records.

2. Unit vs dimension — the crucial split

Before going further we must separate two ideas people constantly confuse.

We need this distinction because dimensional analysis works at the level of kinds, so it is immune to which unit you happened to pick. That is precisely why it is so robust.

3. The notation $[Q]$ — "the dimension of $Q$"

Read it as a question the brackets ask: "what colour is this?" For example:

• $[\text{a distance}]=L$
• $[\text{a duration}]=T$
• $[\text{a heartbeat count}]=M^0{L}^0{T}^0$ (a pure number — no colour at all)

That last line introduces the most important habit of the whole topic: a pure number has no dimension, which we write with all exponents zero.

4. Exponents on dimensions — multiplying and dividing kinds

Here is where symbols like $L^2$ and $T^{-2}$ come from, and they are easy once you see the picture.

The notation $T^{-1}$ simply means "per time" (one division by time). $T^{-2}$ means "per time, per time" — divided by time twice.

In the figure, watch how a length bar (blue) stacks with itself to make an area — two blue bars, hence $L^2$. Below it, a length is cut by a time to give speed: the yellow time sits in the denominator, which is what the negative exponent records.

5. What "$=$" really demands: same-kind-only addition

This single restriction is the Principle of Homogeneity, and it is the engine of the whole parent topic. Everything else — checking equations, deriving the pendulum law — is just this rule applied carefully.

The figure shows two sums. On the left, two blue length-bars add cleanly into a longer blue bar — legal, same colour. On the right, a blue length-bar and a yellow time-bar are shoved together — the result is a question mark, because there is no kind that is both. That crossed-out sum is exactly what dimensional analysis hunts for in a wrong equation.

6. Why $\sin$, $\ln$, $e^x$ need naked numbers

The last symbol-family the parent uses is functions like $\sin$, $\cos$, $\ln$, and the exponential $e^x$. There is a beautiful reason their inputs must be pure numbers.

So whenever you see $\sin (\theta )$ or $e^{-kt}$ in physics, the thing inside is guaranteed dimensionless. That is not a convention; it is forced by the same addition rule.

The prerequisite map

Read top to bottom: knowing what a quantity is lets us name the three base kinds; the bracket notation and exponent rules turn those kinds into algebra; homogeneity is the single rule that powers both checking and deriving; and the function-argument rule is a direct corollary. All arrows converge on the parent topic.

Related vault topics

• Units and the SI system — the sizes (units) that realise these kinds.
• Significant figures and error propagation — handling the numbers once kinds are settled.
• Vectors — components and addition — quantities that also carry direction, each component still wearing a kind.
• Equations of motion (kinematics) — the first equations you'll dimensionally check.
• Buckingham Pi theorem — the grown-up generalisation of the "match the powers" derivation trick.

Equipment checklist

Test yourself — cover the right side.