Lesson V: Unit Conversion and Dimensional Analysis



Unit Conversion and Dimensional Analysis

Introduction

It is often necessary to convert a value from one system of units into another. Sometimes the unit scale needs to be moved up or down in order to give the outside observer the proper perspective. For instance, if I tell a person that I live 2,600,000 inches from Baton Rouge, they may realize I live far away, yet cannot perceive just how far. If I say, however, that I live 41 miles from Baton Rouge, they can much better visualize the scale. Unit conversion may also be necessitated when quantities measured in one system are combined, then reported in a standardized unit which practioners use on a regular basis. To properly convert units, we employ dimensional analysis. We all have an innate ability to perform dimensional analysis, so we will investigate a manner to systematize this.

Example 1: Simple Conversion Factors

Let's ssy for instance that you purchase donuts at a bakery and buy 3 dozen, it is apparent that you bought 36 donuts. If you want one donut for each of 24 people, you would purchase 2 dozen. In your mind, you were able to make the conversion from the unit of dozen into individual donuts and vice versa. You were also naturally know that you multiply by 12 in the former case, and divide by 12 in the latter. To systematize this process, we will use a conversion factor

1 dozen = 12 donuts

The conversion factor is placed in the form of a fraction, arranged so that the unit to be changed is in the denominator and the unit desired is in the numerator. The units are arranged to algebraically cancel, and this automatically indicates the operation of the numeric values :

Example 2: Multi-Dimensional Conversion

When the unit of measurement is multidimensional, the conversion factor is simply applied a corresponding multiple of times. For instance the area of a rectangular body is found by taking the length times width. Therefore the units associated with area are square inches, square meters, square yards, etc:

Area = 9.620 yd î 2.000 yd = 18.62 yd2

If one wishes to convert this quantity to square feet, you must apply the conversion factor of 1 yd = 3 ft twice:
Equivalently, the conversion factor may be combined directly to create a new conversion factor which converts directly between square yards and square feet:

Example 3: Fractional Unit Conversion

Often units appear in both the numerator and denominator of an expression. For exzmple, the speed of an automobile is measured in MPH or miles/hour. If one wishes to convert this quantity to inches/second, conversion factors must be applied to both the numerator and the denominator:


In the following example, the density of a substance is converted from units of grams/milliliter into pounds/inch3. Notice that this example applies the conversion factor 1 in = 2.54 cm three times:

Unit Conversion and Significant Figures

When converting units, concern must be given to the number of Significant Figures which result from the conversion. Generally, when converting within the same unit systems, conversions are exact, and therefore infinitely significant. For example 1 ft = 12 in, 1 m = 1000 mm, 1 gal = 4 qt are all examples of exact conversions, and therefore do not limit the number of significant figures in a resulting calculation. However, when converting between unit systems many times the conversion factor has limited significance. For example 1.00 lb = 454 g, 1.00 atm = 760 mm Hg, and 1.00 yd = .914 m all only contain three significant figures. Accordingly, when they are used in unit conversion the result can at most contain only three significant figures.



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