Dimensional Analysis

The dimensions of a quantity describe the type of quantity being measured. For example, any distance or length, regardless of units, has the dimension of length, which has the symbol . This can be written as .

There are three base dimensions in mechanics:

Length, denoted
Mass, denoted
Time, denoted

A scalar quantity has only magnitude (e.g. speed), whereas a vector quantity has both magnitude and direction (e.g. velocity). Vector quantities have the same dimension as their scalar equivalents (e.g. )

When adding and subtracting quantities, only quantities with the same dimensions can be added or subtracted. The resulting sum or difference has the same dimensions. For products or quotients the dimensions are multiplied or divided.

A dimensionless quantity has dimensions of , where all of the dimensions cancel out.

Many coefficients, such as the coefficient of restitution, are dimensionless.
Pure numbers, such as and in , are dimensionless.
Angles have units but are dimensionless.
Trigonometric functions are dimensionless as they are the ratio of two quantities with the same dimensions.

Dimensional analysis can be used to predict formulae, by equating the dimensions on both sides of a proposed formula. Dimensional analysis can also be used as an error check to see if a formula makes physical sense.

The author advises the reader to complete practice questions to solidify their understanding of this topic.