FHSST Physics/Units/How to Change

How to Change Units-- the "Multiply by 1" Technique

Also known as fractional dimensional analysis, the technique involves multiplying a labeled quantity by a conversion ratio, or knowledge of conversion factors. First, a relationship between the two units that you wish to convert between must be found. Here's a simple example: converting millimetres (mm) to metres (m)-- the SI unit of length. We know that there are 1000 mm in 1 m which we can write as

$\begin{array}{c}1000\text{ mm}=1\text{ m}\end{array}$

Now multiplying both sides by

$\frac{1}{1000\text{ mm}}$

we get

$\begin{array}{c}\frac{1}{1000mm}1000mm=\frac{1}{1000mm}1m,\end{array}$

which simply gives us

$\begin{array}{c}1=\frac{1m}{1000mm}.\end{array}$

This is the conversion ratio from millimetres to metres. You can derive any conversion ratio in this way from a known relationship between two units. Let's use the conversion ratio we have just derived in an example:

Question: Express 3800 mm in metres.

Answer:

$\begin{array}{ccc}3800mm& =& 3800mm\times 1\\ & =& 3800mm\times \frac{1m}{1000mm}\\ & =& 3.8m\end{array}$

Note that we wrote every unit in each step of the calculation. By writing them in and cancelling them properly, we can check that we have the right units when we are finished. We started with mm and multiplied by $\frac{m}{mm}$

This cancelled the mm leaving us with just m, which is the SI unit we wanted! If we wished to do the reverse and convert metres to millimetres, then we would need a conversion ratio with millimetres on the top and metres on the bottom.

It is helpful to understand that units cancel when one is in the numerator and the other is in the denominator. If the unit you are trying to cancel is on the top, then the conversion factor that you multiply it with must be on the bottom.

This same technique can be used to not just to convert units, but can also be used as a way to solve for an unknown quantity. For example: If I was driving at 65 miles per hour, then I could find how far I would go in 5 hours by using $\frac{65\text{ miles}}{1\text{ hour}}$ as a conversion factor.

This would look like $\frac{5\text{ hours}}{1}\frac{65\text{ miles}}{1\text{ hour}}$

This would yield a result of 325 miles because the hours would cancel leaving miles as the only unit.

Practice Problem

Problem: Convert 3 millennia into seconds.

Most people don't know how many seconds are in a millennium, but they do know enough to solve this problem. Since we know 1000 years = 1 millennium, 1 year = about 365.2425 days, 1 day = 24 hours, and 1 hour = 3600 seconds we can solve this problem by multiplying by one many times.

$\frac{3\text{ millennia}}{1}\frac{1000\text{ years}}{1\text{ millennium}}\frac{365.2425\text{ days}}{1\text{ year}}\frac{24\text{ hours}}{1\text{ day}}\frac{3600\text{ seconds}}{1\text{ hour}}$