Using Dimensions

A quick test that catches wrong formulas, converts units and even guesses laws.

13 minUpdated 21 October 2026principle of homogeneityunit conversiondimensional analysislimitations of dimensional analysis

Lesson at a glanceShow the key points

Principle of homogeneity: every term added, subtracted or equated must have the same dimensions.
A dimensionally wrong equation must be wrong; a dimensionally correct one may still be wrong (numbers like ½ are invisible).
Conversion: if [Q] = MᵃLᵇTᶜ, one unit of Q in system 1 = (M₁/M₂)ᵃ(L₁/L₂)ᵇ(T₁/T₂)ᶜ units in system 2; e.g. 1 Pa = 10 CGS units, 1 J = 10⁷ erg.
If a quantity is a product of powers of known quantities, matching dimensions can find the powers: the pendulum period T = k√(l/g).
Limits: the method cannot find dimensionless constants, fails for sums (like ut + ½at²) and needs no more unknown powers than independent dimensions (three in mechanics).

Dimensions look like bookkeeping, but they are a pocket tool: they catch wrong formulas in seconds, convert units between systems, and can even guess the form of a law you haven’t derived.

1. Checking equations

You can add apples to apples but not apples to kilometres. In physics, you can only add, subtract or equate quantities of the same kind:

Principle of homogeneity

\text{Every term in a physical equation has the same dimensions.}

Check $v^2 = u^2 + 2ax$ , for a particle accelerating uniformly from speed u through distance x:

[v^2] = [u^2] = \mathrm{L^2T^{-2}}, \qquad [2ax] = \mathrm{LT^{-2}}\cdot\mathrm{L} = \mathrm{L^2T^{-2}} \ \checkmark

Interactive

Is this equation possible?

Pick an equation. Each term's dimensions are worked out; if any term disagrees with the others, the equation cannot be right.

x = ut + \tfrac12at^2

$x$ $[\,] = L$
$ut$ $[\,] = LT^{-1}\cdot T = L$
$\tfrac12at^2$ $[\,] = LT^{-2}\cdot T^2 = L$

Every term is a length: dimensionally consistent.

verdictdimensionally consistent

Equation

Worked example

Could this be right?

A student claims the speed of a wave on a stretched string is v = √(F/μ), where F is the tension and μ the mass per unit length. Check it dimensionally.

2. Converting units

Change the units of mass, length and time, and every derived unit changes with them. If a quantity has dimensions $\mathrm{M^aL^bT^c}$ , then

Conversion factor

\frac{1\ \text{unit in system 1}}{1\ \text{unit in system 2}} = \Big(\frac{M_1}{M_2}\Big)^a\Big(\frac{L_1}{L_2}\Big)^b\Big(\frac{T_1}{T_2}\Big)^c

Worked example

Joules into ergs

The CGS system uses the gram, centimetre and second; its unit of energy is the erg. How many ergs make one joule?

Interactive

Converting a derived unit

Write the quantity's dimensional formula MᵃLᵇTᶜ, then replace each base unit by how many new units it contains. The powers do the rest.

[\text{pressure}] = ML^{-1}T^{-2}

\frac{1\ \text{pascal}}{1\ \text{new unit}} = \Big(\frac{1\ \text{kg}}{1\ \text{g}}\Big)^{1}\Big(\frac{1\ \text{m}}{1\ \text{cm}}\Big)^{-1}\Big(\frac{1\ \text{s}}{1\ \text{s}}\Big)^{-2} = (1)^{1}(100)^{-1}(1)^{-2}

1 pascal =10 g·cm⁻¹·s⁻²

Quantity

New system

If you know which quantities something depends on, and suspect the dependence is a product of powers, dimensions can find the powers. What sets the speed v of a wave on a string? Plausibly the tension F and the mass per unit length μ. Try

v = k\,F^a\mu^b

where k is a dimensionless number. Then

\mathrm{LT^{-1}} = (\mathrm{MLT^{-2}})^a(\mathrm{ML^{-1}})^b = \mathrm{M^{a+b}\,L^{a-b}\,T^{-2a}}

Match the powers: M: a + b = 0; L: a − b = 1; T: −2a = −1. So a = ½, b = −½, and

v = k\sqrt{F/\mu}

Worked example

The period of a simple pendulum

The period t of a pendulum might depend on its length l, the bob’s mass m and g. Assuming t = k lᵃ mᵇ gᶜ, find a, b and c.

Limits of the method

You must already know which quantities are involved.
It only works for products of powers. It cannot produce $x = ut + \tfrac12at^2$ , which is a sum of two terms.
It never gives dimensionless constants such as 2π or ½.
You get one equation per base quantity. In mechanics that is three (M, L, T), so at most three unknown powers can be found.

What to remember

1Homogeneity: all terms of an equation must have the same dimensions. Wrong dimensions ⇒ wrong equation, but not vice versa.
2Conversion factor between systems = product of base-unit ratios raised to the dimensions: 1 J = 10⁷ erg.
3Assume a product of powers and match dimensions to find the exponents (e.g. v = k√(F/μ), t = k√(l/g)).
4Limits: needs known variables, products only, no numerical constants, at most three unknowns in mechanics.

Check yourself

An equation reads s = ut + ½at³. The dimension check shows it is…

1 newton equals how many dynes (g·cm/s²)?

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

WatchTreating units algebraically and dimensional analysisKhan Academy WatchUnits, dimensions, uncertaintiesWalter Lewin · MIT 8.01x Lecture 1The famous ‘any measurement without uncertainty is meaningless’ lecture ReadDimensional analysisOpenStax University Physics §1.4 ReadUnit conversionOpenStax University Physics §1.3 ReadDimensional analysisWikipedia

PreviousDimensions Next Orders of Magnitude & Estimation

Using Dimensions

A quick test that catches wrong formulas, converts units and even guesses laws.

13 minUpdated 21 October 2026principle of homogeneityunit conversiondimensional analysislimitations of dimensional analysis

Lesson at a glanceShow the key points

Principle of homogeneity: every term added, subtracted or equated must have the same dimensions.
A dimensionally wrong equation must be wrong; a dimensionally correct one may still be wrong (numbers like ½ are invisible).
Conversion: if [Q] = MᵃLᵇTᶜ, one unit of Q in system 1 = (M₁/M₂)ᵃ(L₁/L₂)ᵇ(T₁/T₂)ᶜ units in system 2; e.g. 1 Pa = 10 CGS units, 1 J = 10⁷ erg.
If a quantity is a product of powers of known quantities, matching dimensions can find the powers: the pendulum period T = k√(l/g).
Limits: the method cannot find dimensionless constants, fails for sums (like ut + ½at²) and needs no more unknown powers than independent dimensions (three in mechanics).

Dimensions look like bookkeeping, but they are a pocket tool: they catch wrong formulas in seconds, convert units between systems, and can even guess the form of a law you haven’t derived.

1. Checking equations

You can add apples to apples but not apples to kilometres. In physics, you can only add, subtract or equate quantities of the same kind:

Principle of homogeneity

\text{Every term in a physical equation has the same dimensions.}

Check $v^2 = u^2 + 2ax$ , for a particle accelerating uniformly from speed u through distance x:

[v^2] = [u^2] = \mathrm{L^2T^{-2}}, \qquad [2ax] = \mathrm{LT^{-2}}\cdot\mathrm{L} = \mathrm{L^2T^{-2}} \ \checkmark

Interactive

Is this equation possible?

Pick an equation. Each term's dimensions are worked out; if any term disagrees with the others, the equation cannot be right.

x = ut + \tfrac12at^2

$x$ $[\,] = L$
$ut$ $[\,] = LT^{-1}\cdot T = L$
$\tfrac12at^2$ $[\,] = LT^{-2}\cdot T^2 = L$

Every term is a length: dimensionally consistent.

verdictdimensionally consistent

Equation

Worked example

Could this be right?

A student claims the speed of a wave on a stretched string is v = √(F/μ), where F is the tension and μ the mass per unit length. Check it dimensionally.

2. Converting units

Change the units of mass, length and time, and every derived unit changes with them. If a quantity has dimensions $\mathrm{M^aL^bT^c}$ , then

Conversion factor

\frac{1\ \text{unit in system 1}}{1\ \text{unit in system 2}} = \Big(\frac{M_1}{M_2}\Big)^a\Big(\frac{L_1}{L_2}\Big)^b\Big(\frac{T_1}{T_2}\Big)^c

Worked example

Joules into ergs

The CGS system uses the gram, centimetre and second; its unit of energy is the erg. How many ergs make one joule?

Interactive

Converting a derived unit

Write the quantity's dimensional formula MᵃLᵇTᶜ, then replace each base unit by how many new units it contains. The powers do the rest.

[\text{pressure}] = ML^{-1}T^{-2}

\frac{1\ \text{pascal}}{1\ \text{new unit}} = \Big(\frac{1\ \text{kg}}{1\ \text{g}}\Big)^{1}\Big(\frac{1\ \text{m}}{1\ \text{cm}}\Big)^{-1}\Big(\frac{1\ \text{s}}{1\ \text{s}}\Big)^{-2} = (1)^{1}(100)^{-1}(1)^{-2}

1 pascal =10 g·cm⁻¹·s⁻²

Quantity

New system

3. Guessing the form of a law

v = k\,F^a\mu^b

where k is a dimensionless number. Then

\mathrm{LT^{-1}} = (\mathrm{MLT^{-2}})^a(\mathrm{ML^{-1}})^b = \mathrm{M^{a+b}\,L^{a-b}\,T^{-2a}}

Match the powers: M: a + b = 0; L: a − b = 1; T: −2a = −1. So a = ½, b = −½, and

v = k\sqrt{F/\mu}

Worked example

The period of a simple pendulum

The period t of a pendulum might depend on its length l, the bob’s mass m and g. Assuming t = k lᵃ mᵇ gᶜ, find a, b and c.

Limits of the method

You must already know which quantities are involved.
It only works for products of powers. It cannot produce $x = ut + \tfrac12at^2$ , which is a sum of two terms.
It never gives dimensionless constants such as 2π or ½.
You get one equation per base quantity. In mechanics that is three (M, L, T), so at most three unknown powers can be found.

What to remember

1Homogeneity: all terms of an equation must have the same dimensions. Wrong dimensions ⇒ wrong equation, but not vice versa.
2Conversion factor between systems = product of base-unit ratios raised to the dimensions: 1 J = 10⁷ erg.
3Assume a product of powers and match dimensions to find the exponents (e.g. v = k√(F/μ), t = k√(l/g)).
4Limits: needs known variables, products only, no numerical constants, at most three unknowns in mechanics.

Check yourself

An equation reads s = ut + ½at³. The dimension check shows it is…

1 newton equals how many dynes (g·cm/s²)?

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

PreviousDimensions Next Orders of Magnitude & Estimation

1. Checking equations#

2. Converting units#

3. Guessing the form of a law#

Limits of the method

Check yourself

Keep exploring

1. Checking equations#

2. Converting units#

3. Guessing the form of a law#

Limits of the method

Check yourself

Keep exploring

1. Checking equations

2. Converting units

3. Guessing the form of a law

1. Checking equations

2. Converting units

3. Guessing the form of a law