Dimensions of physical quantities

Physics · Measurements. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

Test this topic free All Physics topics

The idea

The dimensions of a quantity show how it is built from mass [M], length [L] and time [T], whatever units are used. Speed is a length divided by a time, so its dimensions are [LT⁻¹].

An equation can only be right if every term in it has the same dimensions. This is the principle of homogeneity, and it gives a quick check on any formula. The same idea lets us derive the form of a formula: write the quantity as a product of powers of the things it depends on, then compare the powers of M, L and T on both sides.

Numbers, angles, and functions such as sin θ have no dimensions.

Rules to remember

Velocity [LT⁻¹]; acceleration [LT⁻²]; force [MLT⁻²].
Work, energy and torque [ML²T⁻²]; power [ML²T⁻³]; momentum and impulse [MLT⁻¹].
Pressure [ML⁻¹T⁻²]; frequency and angular frequency [T⁻¹].
Homogeneity: dimensions on the left side = dimensions of every term on the right side.
Dimensional analysis cannot find a numerical constant such as 2π or ½.
A dimensionally correct equation may still be wrong; a dimensionally wrong equation is certainly wrong.

Common mistake

Students think that passing the dimension check proves an equation. It only shows the equation is possible, because a missing or wrong number such as ½ does not change the dimensions.

Worked example

The centripetal force F on a body depends on its mass m, its speed v and the radius r of its path. Use dimensions to find the form of the formula.

Let F = constant × m^a v^b r^c, so [MLT⁻²] = [M]^a [LT⁻¹]^b [L]^c.
Compare powers: M gives a = 1; T gives −b = −2, so b = 2.
L gives b + c = 1, so 2 + c = 1 and c = −1.
So F = constant × m v² r⁻¹.

Answer: F = constant × mv²/r (the constant, which is 1, cannot be found by this method)

Ask about this

Type what you are stuck on. The helper finds the explanation, worked examples and the notes to read, free and instant.

Keep going

← Errors and uncertainties Vectors: representation, addition and rectangular components →Take a Physics test