Question

How does dimensional analysis work?

Answer 1

Dimensional analysis is simply a way of testing whether the base units of a given equation work out. It operates on a simple principle: the units you have on one side of an equation must match those that you have on the other.

It's analogous to baking a cake. Your cake can only have what ingredients you put in to it.

Consider the following equation:

`F=ma`

Force in base units is `kg*m/s^2`
Mass is just `kg`
Acceleration is `m/s^2`

So now let's evaluate this by plugging in these units into the equation:

`kg*m/s^2 = kg*m/s^2`

This equation works, therefore it can be classified as dimensionally correct.

Let's look at another equation:

`Deltax=v_ot+1/2a^2t^2`

`Deltax` is a distance, measured in meters (`m`)
`v_o` is a velocity, measured in meters per second (`m/s`)
`a` is acceleration, measured in meters per second squared (`m/s^2`)
`t` is time, measured in seconds (`s`)

Now let's just plug everything in:

`m = m/s*s+(m/s^2)^2*s^2`

Notice that I have not included the `1/2`. This is because coefficients do not matter in dimensional analysis because they don't really change the dimension (i.e. half a mass is still a mass).

Now we simplify:

`m = m+(m^2/s^4)s^2`

`m = m+(m^2/s^2)`

Since this equation does not work out, this equation is not dimensionally correct.

Hope that helped :)