Section 4.11 Complicated Conversions
Units as Ratios
In the previous example (and a lot of the time in engineering applications), conversions can be completed in one step and with one conversion factor. Sometimes, instances will arise where you need to perform multiple steps and use multiple conversion factors. As long as you understand how to use the steps for converting units, you can apply that logic to more complicated situations.
For example, lets consider converting \(\frac{55\,miles}{hour}\) to \(\frac{kilometers}{second}\text{.}\) In this case, we can see that we are going to need to convert miles to kilometers and hours to seconds. Although this seems more complicated, it is almost the exact same conversion process as we have done before. This time though, we need to do 2 conversions. See if you can follow along with the steps we outlined above:
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Determine the appropriate conversion factor(s) to be used.In this case, the conversion factors of interest will be:\(1\,kilometer=0.621\,miles\) AND \(1\,hour=3600\,seconds\)
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Set up a ratio based on the conversion factor that will result in the current unit being eliminated, and the unit of interest being preserved.\(\frac{1\,kilometer}{0.621\,miles}\) AND \(\frac{1\,hour}{3600\,seconds}\)We set up the ratios this way so that when we multiply \(55\,\frac{miles}{hour}\) by those ratios, we can see that miles will be eliminated and kilometers will remain in the numerator AND hour will be eliminated but seconds will remain in the denominator.
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Perform multiplication and algebra.Multiplying everything and canceling reveals:\(55\,\frac{miles}{hour}*\frac{1\,kilometer}{0.621\,miles}*\frac{1\,hour}{3600\,seconds}=0.0246\,\frac{kilometers}{seconds}\)
Units with Exponents
The last thing you need to be aware of when you are performing unit conversions is when we look at squared, cubed, and other higher power units. For example, lets consider the dimension volume. We know that SI unit for the dimension volume is the cubic meter (\(m^3\)). The important thing here is the “cubic” part. If we want to convert from cubic meters to say, cubic inches, we can use the conversion factors: \(1\,meter=3.281\,feet\) and \(1\,foot=12\.inches\) in a two-part process to eliminate meters and end up with inches. Notice, however, that our conversion factors are NOT in powers of three. We have to account for that when we are performing our conversions. All you have to do, is cube the ratios that you develop during Step 2 of our conversion process technique.
To illustrate this, lets convert \(12.3\,m^3\) to \(in^3\text{.}\) We will follow the same steps we used in the previous examples.
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Determine the appropriate conversion factor(s) to be used.In this case, we already discussed that the conversion factors would be:\(1\,meter=3.281\,feet\) AND \(1\,foot=12\,inches\)
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Set up a ratio based on the conversion factor that will result in the current unit being eliminated, and the unit of interest being preserved.This is where things get tricky. In order for this to work, we can tell that we need cubic feet in the numerator for the first multiplication step and cubic meters in the denominator. However, our conversion factors identified do not correspond to the cubic quantities. The solution is simply to cube the appropriate ratio to create a new conversion factor. For this example the two conversion ratios then become:\(\frac{3.281\,ft}{1\,m}*\frac{3.281\,ft}{1\,m}*\frac{3.281\,ft}{1\,m}=\frac{35.32\,ft^3}{1\,m^3}\) AND \(\frac{12\,in}{1\,ft}*\frac{12\,in}{1\,ft}*\frac{12\,in}{1\,ft}=\frac{1728\,in^3}{1\,ft^3}\)
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Perform multiplication and algebra.Now that we have conversion factors with appropriate units, we can perform the multiplication steps and eliminate unwanted units.\(12.3\,m^3*\frac{35.32\,ft^3}{1\,m^3}*\frac{1728\,in^3}{1\,ft^3}=750705\,in^3\)
Checkpoint 4.11.2. Complicated Conversions (Area).
Checkpoint 4.11.3. The Smoot.
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