Dimensional analysis checks whether a physics equation can be correct by comparing the dimensions (mass M, length L, time T) on both sides. If the dimensions do not match, the equation is wrong. It can also derive a relation between quantities, but it cannot find dimensionless constants such as ½ or 2π.
Adding metres to seconds is like adding rice to rupees at a kiryana store: the numbers may look fine, but the answer means nothing. Dimensional analysis catches exactly that kind of mistake.
What are dimensions?
The dimensions of a quantity show how it is built from the base quantities. In mechanics we use mass [M], length [L] and time [T]:
| Quantity | Formula | Dimensions |
|---|---|---|
| Velocity | distance ÷ time | [LT⁻¹] |
| Acceleration | velocity ÷ time | [LT⁻²] |
| Force | mass × acceleration | [MLT⁻²] |
| Work, energy | force × distance | [ML²T⁻²] |
| Power | work ÷ time | [ML²T⁻³] |
Pure numbers such as ½, 2 and π, and angles, have no dimensions.
Use 1: checking an equation
The principle of homogeneity: every term in a correct physical equation has the same dimensions.
Check s = vᵢt + ½at²:
- Left side: s is a length → [L].
- vᵢt: [LT⁻¹] × [T] = [L].
- ½at²: [LT⁻²] × [T²] = [L] (the ½ has no dimensions).
All terms are [L], so the equation is dimensionally correct.
Now check the speed of a wave on a stretched string, v = √(Fl/m), where F is tension, l is length and m is mass:
- Fl/m = [MLT⁻²][L] ÷ [M] = [L²T⁻²]
- √[L²T⁻²] = [LT⁻¹], which is the dimension of speed. ✓
Use 2: deriving a relation (the simple pendulum)
Suppose the time period T of a pendulum depends on its length l and on g. Write T = k lᵃ gᵇ, where k is a pure number.
- Dimensions: [T] = [L]ᵃ [LT⁻²]ᵇ = [L^(a+b) T^(−2b)]
- Compare powers of T: −2b = 1, so b = −½
- Compare powers of L: a + b = 0, so a = ½
So T = k √(l/g). Experiment gives k = 2π, so T = 2π√(l/g). Dimensional analysis found the shape of the formula but not the 2π.
Limitations
- It cannot find dimensionless constants (the 2π above, or the ½ in ½at²).
- It cannot tell you whether a term should be added or subtracted.
- It fails when a quantity depends on a trigonometric, logarithmic or exponential function.
- A dimensionally correct equation can still be wrong: v = u + 2at passes the test but is false.
Common mistakes in the exam
- Giving ½ or π a dimension. Pure numbers are dimensionless.
- Saying "correct dimensions means correct equation". It only means it may be correct.
- Mixing units and dimensions. m s⁻¹ is a unit; [LT⁻¹] is the dimension.
Quick revision
- Dimensions: powers of [M], [L], [T].
- Every term in a correct equation has the same dimensions.
- Derive by matching powers; constants come from experiment.
Practise with an interactive dimension builder, model answers and MCQs in the free lesson 1.8 Dimensions.