How do you calculate dimensions?

Answer 1

When calculating dimensions, you look at the measurement units and ignore the numbers associated with them.

Dimensions are represented in square brackets.

[L] is a single dimension in length, [T] is time and [M] is mass.

[L2] represents length in 2 dimensions - or an area.

Slightly more complex are density = [M][L-3] or [M]/[L3]

Addition of subtraction can only be carried out on identical dimensions and the result is the same "term".

Thus [L2] + [L2] = [L2]. The rules for combination may look strange mathematically, but if you describe the equation in words they may be clear. Add an area to an area and you get and area.

Multiplication and division are "normal" and, if you are familiar with indices, they are straightforward.

Here is an interesting example of where dimensional analysis can take you.

Velocity = [L][T-1]

Acceleration = (Change in velocity)/Time = ([L][T-1] - [L][T-1]) / [T]

but [X] - [X] = [X] (see rules for addition and subtraction above)

= ([L][T-1]) /[T] = [L][T-2]

Force = Mass*Acceleration = [M]*[L][T-2] = [M][L][T-2]

Energy or mechanical Work = Force*Distance Moved = [M][L][T-2]*[L]

= [M][L2][T-2]

= [M]*([L][T-1])2

But [L][T-1] is the dimensional representation of velocity

So, we have

Energy = [M]*velocity2

Or e =mc2 which you may have come across before!