Unit Conversions

How to do unit conversions in Physics is simpler but essential. The unit conversions examples here should be more than enough for any unit conversions.

We often need to convert units from one standard to another for example, mile to meter, hour to second, meter to inch, feet to meter, kilogram to gram etc. You need to know the length equivalent of meter to inch or kilometer to mile etc. such as $1\text{m}=39.37\text{in}$ or $1\text{km}=0.6214\text{mi}$.

There are various standards of units and we are considering only the SI units and you need to know the equivalent of SI units to other units to convert the units to the SI system. For example you are currently converting $2.5\text{hr}$ to second. The easy and memorable way to do this is multiplying $2.5\text{hr}$ by $\frac{3600\text{s}}{1\text{hr}}$ since you already know $1\text{hr}=3600\text{s}$.

Notice that $\frac{3600\text{s}}{1\text{hr}}$ is $1$ and multiplying by $1$ doesn't change the value. It's easier to remember "multiply by one" while converting units. You can cancel or multiply units and determine whether the units are consistent or not on both sides of an equation. You can treat units like algebraic quantities and cancel them like dimensions.

2.5 hr ( 3600s 1 hr )=9000s

Another example is to convert $45\text{km}$ to mile. You may know $1\text{mi}=1.609\text{km}$. Now we multiply $45\text{km}$ by $\frac{1\text{mi}}{1.609\text{km}}$ in order to cancel $\text{km}$. Note that multiplying by $\frac{1\text{mi}}{1.609\text{km}}$ is the same thing as multiplying by $1$ as you already know $1\text{mi}=1.609\text{km}$.

45.00 km ( 1mi 1.609 km )=27.97mi

You should carry units throughout calculation in problems to see whether the units are consistent or not. It means that if you are finding the speed of a car, the unit of speed should come as $\text{m/s}$ on both side of the equation v=distance/time. We always carry the units through the calculation as it is a good way to know the units are consistent or not in our calculation.

Example 1| Convert one cubic inches to cubic meter.

You should know $1\text{m=39}\text{.37in}$ and $1{{\text{m}}^{3}}={{(9.37\text{in)}}^{3}}$ so we multiply $\text{1i}{{\text{n}}^{3}}$ by the ratio $\frac{1{{\text{m}}^{3}}}{{{(39.37)}^{3}}\text{i}{{\text{n}}^{3}}}$ which is $1$ and multiplying by $1$ does not change the value.

1in 3 = 1 in 3 1 m 3 (39.37) 3 in 3 =1.64× 10 5 m 3

Example 2 | If a car moves at speed 75.0 mi/hr, express the speed in meters per second.

In this case we need to cancel both miles and hours. If you practice more you can do this in one step but I do here in two steps.

First I convert miles to meters and to do that I multiply $75.0 \kern 2pt \text{mi/hr}$ by $\frac{1609 \kern 2pt \text{m}}{1 \kern 2pt \text{mi}}$ as you know $1 \kern 2pt \text{mi} = 1609 \kern 2pt \text{m}$ to cancel miles and it becomes meters per hour. We complete the conversion in the next step.

\[75.0\,{\rm{mi/hr = }}75.0\,\frac{{{\rm{mi}}}}{{{\rm{hr}}}}\left( {\frac{{1609\,{\rm{m}}}}{{1\,{\rm{mi}}}}} \right) = 120675\,{\rm{m/hr}}\]

Now meters per hour is not our result. We need to cancel hours and convert it into second. You apply the same multiply by one rule in such a way that the units which are no longer needed are canceled.

\[120675\,{\rm{m/hr}} = 120675\,\frac{{\rm{m}}}{{{\rm{hr}}}}\left( {\frac{{1\,{\rm{hr}}}}{{60\,\min }}} \right)\left( {\frac{{1\min }}{{60\,{\rm{s}}}}} \right) = 33.5\,{\rm{m/s}}\]

Do a little more practice on this and your unit conversion skill will be blazing fast! Always use the units in your calculation for unit consistency as well as for unit conversions. By carrying out the units in calculation, you can detect your pitfalls if something goes wrong.

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