IB Physics SLTool 3 — MathematicsPaper 1 & 2Checking units balance~7 min read
Dimensional Analysis
Here’s a superpower hiding in plain sight: you can spot a wrong equation without knowing a single number. If the units on the left of an equals sign don’t match the units on the right, the equation simply cannot be true — no amount of clever algebra will save it. Dimensional analysis is the habit of checking that balance, and it catches mistakes, confirms rearrangements, and even proves that a constant has no units at all.
📘 What you need to know
A valid physical equation must be homogeneous — the base units on both sides are identical
To check, rewrite every quantity in SI base units and compare the two sides
If the units don’t match, the equation is wrong and needs adjusting
The method can prove a constant is dimensionless (has no units) if both sides already balance without it
You can only add or subtract quantities with the same units — a length plus a time is meaningless
Matching units is necessary but not sufficient: it can’t spot a wrong pure number or a missing dimensionless factor
Why units must balance
Think of an equation as a set of scales. Whatever quantity sits on the left, an equal quantity must sit on the right — and that includes the type of quantity, not just its size. You can’t balance 5 kg against 5 seconds; they’re different things. So the units on each side have to be the same, or the equation is nonsense.
Both pans carry the same base units, so the beam stays level — the equation is homogeneous.
How to check homogeneity
The routine is short and mechanical. Take each side of the equation in turn, swap every quantity for its base units, simplify, and compare.
🧭 Checking an equation’s units
Write the base units of every quantity in the equation
Simplify the left-hand side to a single combination of base units
Simplify the right-hand side the same way
Compare: identical units → homogeneous; different → the equation is wrong
Units of LHS
→ must equal →
Units of RHS
→ if not →
Equation is wrong
One rule flows straight out of this: you can only add or subtract things with the same units. That’s why a formula like s = ut + ½at² works — every term comes out in metres. If you ever find yourself adding a velocity to an acceleration, stop: something’s gone wrong upstream.
Proving a constant is dimensionless
A neat use of the method is to show that a constant carries no units. If both sides of an equation already balance without the constant contributing anything, then the constant must be a pure number — dimensionless. The speed of sound in a gas gives a classic example.
Speed of sound in a gasv = √(γp ÷ ρ)
Quick recap: a real equation is homogeneous — both sides share the same base units; rewrite each quantity in base units and compare, and only ever add or subtract quantities with matching units.
WE 1
The speed v of sound in a gas is given by v = √(γp ÷ ρ), where p is pressure and ρ is density. Show that the constant γ has no units.
Assume γ is dimensionless, work out the base units of the right-hand side, and check they match the base units of speed.
Base units of each quantity
v = m s⁻¹ ; p = Pa = kg m⁻¹ s⁻² ; ρ = kg m⁻³
Units of p ÷ ρ(kg m⁻¹ s⁻²) ÷ (kg m⁻³) = m² s⁻²Take the square root
√(m² s⁻²) = m s⁻¹
RHS = m s⁻¹ = units of v ✓The two sides already balance without γ adding any units, so γ must be dimensionless.
WE 2
A student proposes that a wave speed is given by v = √(g ÷ L), where g is an acceleration and L is a length. Use dimensional analysis to decide whether this equation could be correct.
Find the base units of the right-hand side and compare them with the units of a speed.
Base units
v should be m s⁻¹ ; g = m s⁻² ; L = m
Units of g ÷ L(m s⁻²) ÷ (m) = s⁻²Square root
√(s⁻²) = s⁻¹
RHS = s⁻¹ ≠ m s⁻¹The units don’t match a speed, so the equation cannot be right. The correct form is v = √(gL), which does give m s⁻¹.
💡 Top tips
Do each side separately, then compare at the end — don’t try to juggle both at once
Square roots halve the powers and squares double them — apply them to every base unit inside
Constants like π, angles, and ratios are dimensionless, so they never affect the unit check
Use it as a fast sanity check on any rearrangement — if the units come out wrong, you slipped somewhere
⚠ Common mistakes
Thinking matching units prove an equation — they only rule out impossible ones; a wrong number still passes
Forgetting to apply a power or root to every unit inside a bracket
Treating a prefixed unit as base — convert kW, cm, ms to base units first
Adding or subtracting terms with different units and not noticing the equation is already broken
Up next: Processing Uncertainties — every measurement carries a margin of doubt, so we’ll look at absolute, fractional and percentage uncertainty, and the rules for combining them when you add, multiply, or raise to a power.
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