The dimensions of a quantity state which base quantities it is made from, and with what powers. Speed is length divided by time whether it is measured in m/s or km/h.
Speed can be measured in metres per second, kilometres per hour, or miles per hour. These units differ in size. But each one is a unit of length divided by a unit of time.
The dimensions of a quantity describe that structure: which base quantities it is made from, and with what powers. Dimensions do not depend on the choice of unit.
The base quantities of mechanics are written as capital letters in square brackets: [M] for mass, [L] for length and [T] for time. Other base quantities have their own symbols, for example [A] for electric current and [K] for temperature.
The dimensional formula of a quantity writes it as a product of powers of these. Speed is length divided by time:
[speed] = [L T⁻¹]
This reads “length to the power 1, time to the power −1”. A negative power means the quantity is in the denominator.
Acceleration is change in velocity divided by time. What is its dimensional formula?
Velocity has dimensions [L T⁻¹]. Dividing by a time multiplies by [T⁻¹]:
[acceleration] = [L T⁻¹] × [T⁻¹] = [L T⁻²]
This matches the SI unit of acceleration, m/s²: metres per second, per second.
To find the dimensions of a quantity, write its defining equation and replace each quantity by its dimensions.
Dimensions from definitions. Each dimensional formula comes from the defining equation. Force and energy differ by one factor of [L], because energy (work) is force multiplied by a distance.
Some quantities have no dimensions. An angle in radians is defined as arc length divided by radius. That is [L] divided by [L], so the dimensions cancel and the angle is a pure number. Any ratio of two quantities of the same kind is dimensionless, as are pure numbers such as ½ and 2π.
This has a consequence for the next session. A dimensional check cannot detect a missing or wrong pure number, because pure numbers have no dimensions to check.
Force has dimensions [M L T⁻²] and energy has [M L² T⁻²]. From the dimensions alone, how are they related?
Energy has one more factor of [L] than force. The [M] and the [T⁻²] are the same.
So energy has the dimensions of force multiplied by a length. This agrees with the definition of work: force multiplied by the distance moved in the direction of the force.
1The dimensions of a quantity describe…
2Using M, L and T, the dimensional formula of velocity is [____].
3What is the dimensional formula of acceleration?
4Force = mass × acceleration. Its dimensional formula is…
5Energy is [M L² T⁻²] and force is [M L T⁻²]. So energy has the dimensions of force times…
6Which quantity is dimensionless?
Next: using dimensions to check whether an equation can be correct.
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