Problems for Intermediate Methods in Theoretical Physics Edward F. Redish

Explaining Dimensional Analysis

Explain why we do dimensional analysis. Note: This is an essay question. Your answer will be judged not solely on its correctness, but for its depth, coherence, and clarity.

Solution

The critical reason that we do dimensional analysis is what we might refer to as "the passive transformation problem." We have to make an arbitrary choice when we assign a dimension to a physical quantity: what standard unit do we use? This "standard unit" is not a part of something belonging to the physical quantity: it is a description we choose to make -- and we could make some other easily. For example, if we are measuring a length, we could choose to measure it in centimeters, inches, furlongs, or light years. We do not want any equality that we write down to depend on an arbitrary choice. We want it to remain true no matter what arbitrary choice we make. We therefore have to insist that our equations are dimensionally consistent: when we change our units, all the terms in the equation must change in the same way so that the truth value is retained.

This condition leads us to a number of useful results.

1. It allows us to check our equations to see that we haven't made an algebraic mistake.
2. It allows us to define combination quantities that have physical significance, such as velocity, momentum, and energy.
But more than that, it allows us to gain some deeper insights into our equations and our solutions.
1. It often allows us to see how our solution will depend on the dimensioned parameters of the problem, allowing us to almost solve (up to a dimensionless constant) a complex problem without actually going through the solution.
2. It often allows us to propose new physical quantities, such as forces, on dimensional arguments alone.
3. It allows us to create "natural scales" for a problem. These may lead to the creation of dimensionless parameters that tell when one phenomenon dominates another. This allows us to propose simplifying approximations.
4. It allows us to reduce (or eliminate) the dimensions in a complex equation by expressing variables in terms of their natural scales. This simplifies the equation for mathematical processing and reduces the number of times we have to solve it (if we are solving numerically).
Many others could be suggested.

[GRADING PATTERN: The first is the deepest and most significant. It was worth 5 pts. If it was not present in some form, the maximum for the problem was 8. The items in the first list are not as significant as those in the second, so first list items were worth 2 pts each, second list items 3 pts each up to a total of 10.]

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Edward F. Redish
Department of Physics
University of Maryland
College Park, MD 20742
Phone: (301) 405-6120
Email: redish@umd.edu

Last revision 16. October, 2005.