In this class, and in others you'll experience during your engineering training, homework and test answers will not be acceptable (and marked WRONG!) unless the answer has the correct units attached. Often, penalties will be assessed unless the correct units are carried through all parts of the calculation.
The insistence on units isn't just a school thing. Recently, a NASA probe failed because of a botched conversion in its software.
To convert a quantity in terms of one unit to an equivalent in new units, multiply by a "conversion factor" --- a ratio of equivalent quantities.
Conversion factors are dimensionless (but not unit-less) and numerically equivalent to unity.
When you apply a conversion factor, the old units cancel out and the new units remain.
You want to do your conversions efficiently -- and spend only the necessary time looking up values. Often it is faster to use several conversion factors that you already know, rather than spend time looking up a single factor that will do the change all at once.
Carrying units through a calculation can be helpful in figuring out a problem. Often, looking at the units provides a clue as to what step needs to be taken next. It also helps prevent silly mistakes, such as accidentally multiplying when you mean to divide.
It is 1953 and your Mercury has been giving you 17 miles to the
gallon. The average cost of gasoline between Baton Rouge and
Bogalusa, a distance of about 100 miles, is 55 cents a gallon.
How much does it cost to make the trip?
I want to point out two mistakes beginners often make when applying conversion factors. First, people will sometimes write out the factor with the numbers upside down. This is usually because they misinterpret the arrangement of the conversion factor table. To avoid this mistake, try to think about the relative size of the units when you write down a factor (i.e. inches are smaller than feet, so the number on the inches side of the line should be bigger).
The other common error has to do with units that are raised to powers, such as "square feet" or "cubic meters". The conversion factors for these need to be raised to the same power as the unit; to go from cubic meters to cubic feet, you use the conversion from meter to foot, but raise everything (numbers and units) to the 3rd power.
I've graded a lot of papers where people didn't apply the power to the number -- so watch this one.
Often, books that are trying to be "practical" present equations and formulae that are unit specific, instead of using more "theoretical" forms that work with any set of units. Handbooks for practicing engineers of such dimensional equations.
For instance, if we want to calculate the power needed for a fan to move a volume rate of air through a pressure difference, the general formula is:
These kinds of dimensional formulas are encountered in many cases. They are convenient when calculations have become standardized and repetitious; however, they can be tricky unless you are careful. They must be used with the units provided. If you look back over how we came up with the equation, you'll note that the numerical constant in the formula has units attached -- even though they aren't written out.
When you come across a formula in a book and units for the variables are specified, you need to be aware that the constants may also have units traveling incognito. For instance, the formula for liquid flow through a pipe can be (very roughly) approximated by:
If you want to use the equation in any way except with the exact given units, you need to examine the equation to determine the hidden units. This isn't hard, but for some of you may take a little practice. For the flow equation, they come out to be
Being able to "back out" the units of an equation is a useful skill, and one that helps with a variety of engineering and thermodynamics problems. It might be a useful exercise to spend some time practicing looking at an equation and thinking of it in terms of units only.
Now suppose that you want to use the flow equation with different units, say m3/second and Pascals. If you only have to do it one time, it isn't hard to convert Pascals to psi, plug the psi into the equation to get gpm, and then convert the gpm to m3/s. But you may be doing an experiment or problem where you need to use the dimensional equation repeatedly. In this case, it makes sense to transform the dimensional equation to the new set of units.
This isn't very difficult. Just remember that the constant (36.5) has all the units attached that make the equation work. All you have to do is convert those attached units:
References:
R.M. Price
Original: 5/25/94
Modified: 6/2/94, 10/20/95, 7/13/98; 5/13/2003; 5/17/2004
Copyright 1998, 2003, 2004 by R.M. Price -- All Rights Reserved
