DIMENSIONS

1. Physical quantities and their dimensions

The dimensions of a physical quantity refer to the type of units that must be used in order to obtain the measure of that quantity. Dimensions are expressed in terms of the base quantities and are denoted as shown below:

The dimensions are not dependent on the actual units used or on the details of the shape of the object whose are is being measured. The formulae for the areas (A) of the triangle and the circle are different but the dimensions of the areas in both cases are the same, [L2]. You may see some examples of physical quantities and their dimensions.

2. Dimensional analysis

Examining the dimensions of physical quantities is useful because it enables one to check whether equations are incorrect. Quantities on both sides of an equality must have the same dimensions.

Only quantities with the same dimensions can be added or subtracted. For example, imagine that you (rather hastily!) derived an equation describing the motion of an acceleration body in terms of its final velocity, vf, its initial velocity, vi, its acceleration, a, and the elapsed time, Δt:

vf = viΔt + aΔt

Can this be correct? Examine the dimensions of each term:

[ LT-1 ] = [ LT-1 ][ T ] + [ LT-2 ][ T]
[ LT-1 ] = [ L ] + [ L ][T-1]                 

The term on the left hand side of the equation does not have the same dimensions as the terms on the right hand side, and therefore it must be wrong.

Note that consistency of dimensions does not guarantee correctness!


3. Additional questions


What is the measure of a quantity?

The measure of a physical quantity is a number equal to the physical quantity divided by the unit.

For example, if you pay R120 for a book, the amount paid is the physical quantity, the unit of currency is the rand (1 R) and the measure is the number 120.


Examples of physical quantities and their dimensions

Quantity Dimensions Quantity Dimensions
length [L] force [MLT-2]
area [L2] pressure [ML-1T-2]
volume [L3] energy, work [ML2T-2]
density [ML-3] power [ML2T-3]
speed [LT-1] electric charge [IT]
acceleration [LT-2] electric potential difference [ML2T-3I-1]
momentum [MLT-1] electric resistance [ML2T-3I-2]