Problem-Solving - Units and Dimensional Analysis

Author(s):
Larry Gladney and Dennis DeTurck

To Measure All Things

Modern science begins with observation and description. The appropriate means of description should allow for little ambiguity in comparison of observations from different people. In many ways, science is more of a driver for a universal'' language than almost any human activity. Although mathematics suffices as a universal language for theoretical purposes, when it comes to experimental measurement, we turn to a globally agreed upon set of standards termed SI or Système International.

The SI units for length, mass, and time are the

• meter
• kilogram
• second.

Combinations of these basic units are used to make up other SI units for quantifying energy, power, speed, force, etc. The SI units all make use of the same set of prefixes for specifying appropriate powers of ten of the basic unit. For example, the centimeter is 1/100 of a meter, a kilometer is 1000 meters, etc. The list of accepted prefixes is below.

Factor Prefix Symbol
1018 exa E
1015 peta P
1012 tera E
109 giga G
106 mega M
103 kilo k
102 hecto h
101 deca da
10-1 deci d
10-2 centi c
10-3 milli m
10-6 micro µ
10-9 nano n
10-12 pico p
10-15 femto f
10-18 atto a

All calculations you do for this class should be in terms of SI units. It will not always be necessary to use SI units to get the correct answer, but all of the formulas we use assume that you are using SI units and are defined with that in mind.

Conversion Between Units

Dealing with units can be annoying when it is necessary to transform from one set to another. You will find below a handy interactive conversion table for doing just that.

In terms of operation, we generally will need to simply multiply through by appropriate conversion factors in a calculation to get to the standard units, for example, suppose we need to find the speed of a car in m/s for a particular problem, but the statement of the problem gives the speed as 30 miles/hour. Then, we perform the following operation by getting the conversion factors from the unit conversion program, then

[30 mi/h][1609 m/1 mi][1 h/3600 s] = 13.5 m/s

Another useful reason for keeping track of units is that they generally give the first hint that a numerical error may have occurred in your calculations. Generally, converting from one unit to another, a technique termed dimensional analysis, is a necessary part of the calculation of an answer to a problem, hence if the units do not come out correctly, you can be sure that there is something not right with your solution. Another thing to watch for is reasonableness of the answer. Physics classes generally provide problems with answers that are actually realizable in real life. Cars do not travel 8000 miles/hour, people can't apply 4 million pounds of force to a door. If the numbers don't look like they fit the size of the unit, check your calculation!

If you haven't done it in awhile, you will need to practice a bit with unit conversions. Let's try the following exercises, using the conversion tables that you can access with the following link.

Problems

1. Ten pounds of flour correspond to how many kilograms of flour?

2. A car driving at 65 miles/hour is travelling at what speed in meters/second?

3. What is the value of X that is necessary to convert A, in units of centimeters/fortnight to B where B is feet/century?

Larry Gladney and Dennis DeTurck, "Problem-Solving - Units and Dimensional Analysis," Loci (November 2004)

JOMA

Journal of Online Mathematics and its Applications