Unit Conversions

We often need to convert units in Physics 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 1m = 39.37in or 1km = 0.6214mi.

Unit conversions are required when the quantity expressed is not in our known format such as we usually want to convert units into SI units (in Physics we always convert units to SI units). Sometimes we may not be familiar with the value of the same physical quantity in a different unit format. Or you want to add the value in a calculation involving another unit.

How to Do Unit Conversions

There are various standards of units and we consider only the SI units in Physics to be consistent anywhere in the universe. In metric unit conversions 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 2.5hr to second. The easy and memorable way to do this is multiplying 2.5hr by 3600s/1hr since you already know 1hr = 3600s.

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 and you can use this unit conversions method in your unit conversions. 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. Simply use the conversion factor (1hr = 3600s) in a way to cancel unwanted unit.

2.5 hr ( 3600s 1 hr )=9000s

Another example is to convert 45km to mile. You may know $1\text{mi}=1.609\text{km}$. Now we multiply 45km by $\frac{1\text{mi}}{1.609\text{km}}$ in order to cancel 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 1mi = 1.609km.

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

Unit Conversions 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

Unit Conversions 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.

Mechanics
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