What is Dimensional Analysis?
Date: 11/26/2001 at 21:20:59 From: Danielle Subject: Dimensional Analysis What is dimensional analysis and how does it work?
Date: 11/28/2001 at 12:00:37 From: Doctor Greenie Subject: Re: Dimensional Analysis Hi, Danielle - Dimensional analysis is a tool that can be used to determine how to manipulate formulas by analyzing the dimensions in the formulas. In the physical world, these units are often units of distance, mass (or weight) and time; but many other units are possible (see, for example, the "fun" problem in the Dr. Math archives for which I have provided a link at the end of this response). One elementary application of dimensional analysis is in the memorization of simple geometric formulas. For a particular example, many students, when beginning to learn formulas related to circles, get confused between the formulas for the circumference and area of a circle: circumference = pi * diameter area = pi * (radius squared) Using dimensional analysis, it is impossible to confuse these two formulas: (1) "pi times diameter" In this formula, "pi" is a pure number, and the diameter is a length. If the diameter is in inches, then "pi times diameter" is in inches; if the diameter is in miles, then "pi times diameter" is in miles. The units on "pi times diameter" are always units of length. So "pi times diameter" measures a length - so it can't be an area formula; in particular, it can't be the formula for the area of a circle, But it can be a formula for a length; and in fact it is the formula for the circumference of a circle. (2) "pi * (radius squared)" In this formula, "pi" is again a pure number, and the radius is a length. If the radius is in inches, then "pi * (radius squared)" is in square inches; if the radius is in miles, then "pi * (radius squared)" is in square miles. The units on "pi * (radius squared)" are always units of (length squared), which are units of area. So "pi * (radius squared)" measures an area - so it can't be the formula for any length; in particular, it can't be the formula for the circumference of the circle. But it can be the formula for an area; and in fact it is the formula for the area of a circle. Another elementary application of dimensional analysis is in converting units of measure. For example, young students often get confused over whether to divide by 12 or multiply by 12 when converting a measurement in inches to a measurement in feet: 180 inches = ??? feet You can analyze this problem using dimensional analysis to determine whether to multiply or divide by 12. In this application of dimensional analysis, you make fractions equivalent to "1" and "multiply" your given measurement by those fractions; when you do this, you can "cancel" like units in the numerator and denominator, just as you can cancel common numerical factors in the numerator and denominator of any fraction. In this example, 1 foot = 12 inches, so the following two fractions are equivalent to "1": 1 foot (1) --------- = 1 12 inches and 12 inches (2) --------- = 1 1 foot Now try multiplying the given measurement, "180 inches," by each of these fractions: (1) 1 foot (180 * 1) (feet * inches) 180 inches * --------- = ------------------------- 12 inches 12 inches Here, the units "inches" cancel in numerator and denominator, leaving an answer in feet: 1 foot 180 180 inches * --------- = --- feet = 15 feet 12 inches 12 or (2) 12 inches (180 * 12) (inches * inches) 180 inches * --------- = ---------------------------- = ????!!! 1 foot feet Here, there are no common units in numerator and denominator that can be cancelled - so we know this is not the right way to convert the measurement "180 inches" to feet. Finally, here is a link to a page in the Dr. Math archives where dimensional analysis is used to solve a "fun" problem: Dimensional Analysis http://mathforum.com/library/drmath/view/56728.html I hope this helps. Write back if you have further questions. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/
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