
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.
 It allows us to check our equations to see that we haven't made an algebraic mistake.
 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.
 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.
 It often allows us to propose new physical quantities, such as forces, on dimensional arguments alone.
 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.
 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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This page prepared by
Edward F. Redish
Department of Physics
University of Maryland
College Park, MD 20742
Phone: (301) 4056120
Email: redish@umd.edu
Last revision 16. October, 2005.