Resource Lesson
Dimensional Analysis
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In physics, the
dimension of a quantity
means its type of measure, not a specific unit of measurement. For example, the volume of an object can be measured in liters, cubic meters, and cubic centimeters. Or, we can talk about volume having a dimension of [L]
^{3}
and not specify any one particular system.
Dimensional analysis
, in which you substitute for each variable in a formula its dimension, is a valuable tool that allows you to check whether the powers, products, and quotients are correct in a formula. It does NOT check for whether the numerical coefficients or +/- signs are correct.
To check a formula for
dimensional consistency
, merely substitute for each variable its appropriate dimension value and simply each term. Remember, that you are not actually adding together "quantities" only their "dimensions."
For example, you have forgotten which of the following formulas is the correct one for relating distance traveled, time interval and rate of speed. Is it d = r/t or t = d/r.
We can use dimensional analysis to determine which version of the formula is correct. For distance, d, we use [L]; for time, t, we use [T]; for rate or speed, r, we use [L]/[T].
Now which one works?
d = r/t
t = d/r
[L] = [L]/[T]/[T]
[L] = [L]/[T]
^{2}
No!
[T] = [L]/[L]/[T]
[T] = [T]
Therefore we now know that the formula t = d/r which can be restated as d = rt is dimensionally consistent.
Two further examples of dimensional analysis proofs follow:
Notice in this example,
(m - m) = m, NOT, 0
Notice in this example,
m + m = m, NOT, 2m.
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