Dimensional Analysis

Every quantity has a unit and a dimension. The dimension is usually denoted by one or more capital letters in angular brackets. For example, masses have units of kg and a dimension of [M].

In order to clarify the difference between units and dimensions, let us look at a little less trivial example

• The height of a building can be specified by giving a number and a unit. A typical two-story house is about 8 m tall. In physics we would say h = 8 m.
• You could also specify the height of the house as "800 cm" or "26 and a quarter feet " or h = 800 cm.
• The dimension of the height of the house is length, [L]. The reason that you can specify the height of the house in meters or in feet is that the height and both units have the same dimensionality.
• This may seem quite obvious, but you cannot specify the height of a house in seconds, because seconds have the dimensionality of time, [T].

Rule:

Units that have the same dimensionality can be converted into each other; units with different dimensionality cannot.
All secondary units have dimensions based on the base units:

mass [M], length [L], and time [T].

An equation must have the same dimensionality on both sides.

v = x/t

is an equation, equating the velocity (v) to the quotient of displacement (x) and time (t). The dimension of displacement is length [L] and time is [T]. The dimensionality of velocity, v, is therefore [L]/[T].

© MultiMedia Physics, 1999