Dimensions and dimensional analysis

The dimensions of a quantity show which base quantities it is made of, and to what power. Write mass as [M], length as [L] and time as [T]:

• Area = length × breadth: [L²]. Volume of any shape, a box, a cylinder (πr²h) or a cone: [L³].
• Density = mass/volume: [ML⁻³]. Speed: [LT⁻¹]. Acceleration: [LT⁻²].
• Force = mass × acceleration: [MLT⁻²]. Work and energy (force × distance, ½mv², mgh): [ML²T⁻²]. Pressure = force/area: [ML⁻¹T⁻²].

Quantities with the same dimensions are of the same kind and can be compared, but they are not always the same thing: torque and work both have [ML²T⁻²].

Some quantities have no dimensions at all: an angle (arc/radius), a strain (change in length/length), and anything inside sin, cos, log or exp.

The principle of homogeneity: you can only add, subtract or equate quantities with the same dimensions. You cannot add one metre to one rupee. So check each term:

• Is s = ut² + ½at² right? s is [L], but ut² is [LT⁻¹][T²] = [LT]. The terms do not match, so the equation must be wrong.
• In (P + a/V²)(V − b) = RT, a/V² must be a pressure, so a has the dimensions of PV², [ML⁵T⁻²]; and b must be a volume, [L³].

Dimensions can also build a relation. Suppose the force on a body moving in a circle depends on its mass m, speed v and the radius r: F = k mᵃvᵇrᶜ. Then [MLT⁻²] = [M]ᵃ[LT⁻¹]ᵇ[L]ᶜ. Matching powers: M gives a = 1, T gives b = 2, L gives b + c = 1, so c = −1. Hence F = k mv²/r. The same method gives a pendulum's period T = k√(l/g).

Two limits: a dimensionally right equation may still be wrong by a pure number, and the method can never find the number k (an experiment shows k = 1 for the circle and 2π for the pendulum).