Question

Comparing two sides of an equation to prove both has the same base SI units?

The intensity of a sound wave passing through air is given by

`I = Kvpf^2A^2`

where `I` is the intensity ( power per unit area ),
`K` is a constant without units,
`v` is the speed of sound,
`p` is the density of air,
`f` is the frequency of the wave
and `A` is the amplitude of the wave.

Show that both sides of the equation have the same SI base units.

Answer 1

See process below!

Explanation:

The process of verifying if your equation has the right units on the right and the left us called dimensional analysis, and is a very useful way of checking if your answers to more complex science problems make sense.

The first thing I always do in these kinds of problems is simply break down everything into standard SI units. Let's do this now:

`I` (as it says above) is power per unit area. Power is in units of `kg * m^2/s^3`, and area is simply `m^2`, so we'd have:

`I = (kg * m^2/s^3)/(m^2)`

`K` is a dimensionless constant, and hence can just be ignored in this context.

`v` is a speed, so I'm thinking it has units of `m/s`.

`p` is a density, which has units of mass (`kg`) over volume (`m^3`). This leaves:

`p = (kg)/m^3`

You'll usually use liters or something of that form for most calculations, but if we're making everything SI units it's a good idea to keep it as `m^3`.

`f` is frequency, measured in hertz, or `1/s`.

Lastly, `A` is an amplitude, measured in `m`.

Now, let's plug all of the above into our original equation:

`I = Kvpf^2A^2`
`=> (kg * m^2/s^3)/(m^2) = (m/s)((kg)/m^3)(1/s)^2(m)^2`

Now is the fun part: go in and cancel out everything that divides out!

`=> (kg * cancel(m^2)/s^3)/(cancel(m^2)) = (cancel(m)/s)((kg)/cancel(m^3))(1/s)^2cancel(m^2)`

`=> (kg)/s^3 = (kg)/s^3`

These dimensions match up, so we are good!

In these kinds of problems the final answer is not the big deal -- it's the process of plugging in those base units and seeing how they cancel out. Make sure you master this -- it will come in handy across science disciplines.

Hope that helped :)