Dimensions of physical quantities

Many units, one kind

A speed can be in m/s, km/h or miles per hour. The units differ, but every speed is the same kind of thing: a length divided by a time. That kind is its dimension.

Dimensions

We take the base quantities of mechanics as mass $[\text{M}]$, length $[\text{L}]$ and time $[\text{T}]$. (With current, temperature and the others, we would add $[\text{A}]$, $[\text{K}]$ and so on.)

The dimensions of a quantity are the powers to which these base quantities are raised to make it. Speed is a length over a time:

The expression in square brackets is the dimensional formula. A power of zero is usually left out: $[\text{L}\,\text{T}^{-1}]$.

How to find dimensions

Write any formula for the quantity, and treat the dimensions like algebra: multiplying adds the powers, dividing subtracts them, and a pure number like $2$ or $\pi$ has no dimensions.

What a dimension really says

A dimension tells you how a quantity changes when you change the units, or when you scale a body. Make every length twice as big: an area grows four times, because it is $[\text{L}^2]$. See it.

You should find that the power of $k$ is the power of $\text{L}$: $1$ for a side, $2$ for an area, $3$ for a volume.

Kinds of quantities

• With dimensions, like speed and force.
• Dimensionless: a ratio of two quantities of the same kind. Strain (change in length over length), refractive index (speed over speed), and an angle (arc over radius) are pure numbers. An angle in radians has a unit name but no dimensions.
• Same dimensions, different meaning. Work and torque are both $[\text{M}\,\text{L}^2\,\text{T}^{-2}]$. Pressure and stress share $[\text{M}\,\text{L}^{-1}\,\text{T}^{-2}]$. Dimensions cannot tell them apart.

Watch out. A constant is not always dimensionless. The gravitational constant $G$ has dimensions $[\text{M}^{-1}\,\text{L}^3\,\text{T}^{-2}]$. Only a pure number, like $2\pi$, has none.

A worked example

Find the dimensions of the kinetic energy $\frac{1}{2}mv^2$, and check that it matches the dimensions of work.

1. The $\frac{1}{2}$. A pure number: no dimensions.
2. $mv^2$. $[\text{M}] \times [\text{L}\,\text{T}^{-1}]^2 = [\text{M}\,\text{L}^2\,\text{T}^{-2}]$.
3. Work, $Fs$: $[\text{M}\,\text{L}\,\text{T}^{-2}][\text{L}] = [\text{M}\,\text{L}^2\,\text{T}^{-2}]$. The same, as it must be: kinetic energy is the work done to speed a body up.

Practise with new numbers

Try it in Python

The program stores each dimension as three powers, and builds quantities from their formulas by adding and subtracting the powers.

Remember these

• Dimensions: the powers of $[\text{M}]$, $[\text{L}]$, $[\text{T}]$ in a quantity: $[v] = [\text{L}\,\text{T}^{-1}]$.
• Find them from any formula: multiply adds powers, divide subtracts, pure numbers drop out.
• Ratios of like quantities (angle, strain, refractive index) have no dimensions.
• Different quantities can share dimensions (work and torque); constants can have dimensions ($G$).

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find the dimensions of momentum, $mv$.

2. Find the dimensions of the spring constant $k$ in $F = kx$, where $x$ is an extension.

3. Does an angle measured in degrees have dimensions?

4. A balloon swells until every length is $1.5$ times as big. By how many times does its volume grow?