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{\LARGE Dimensional Analysis}\\

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{\bf Step 1:} Define the problem. First identify the result that you
are looking for (call this $A$ for answer). Then identify a set of
variables $v_i$ that fully constrain the problem.  Specifically, be sure
that if you know the values of all of these variables, then the answer
that you are looking for is fully determined. Next, make sure that
your variables are all independent of each other.

{\bf Step 2:} From your list of variables $v_i$, form a quantity with the
same units as $A$, the result that you seek. Call this $Q$.

{\bf Step 3:} Form as many dimensionless quantities from your list of
variables as you can. For instance, if your variables include a height
and a length, then their ratio is a dimensionless number. If your list
includes a height, width, and length, then you have two independent
dimensionless quantities. Call these quantities $\phi_i$.

{\bf Step 4:} Multiply the quantity from Step 2 times the most general
undetermined function $f$ of your set of $\phi_i$: $A=Q f(\phi_i)$. In
the case where no dimensionless quantities can be formed from your
variables (Step 3), then $f$ is simply an undetermined constant.
Finally, note that although dimensional analysis is a useful
technique, it always gives less information that actually solving the
problem.


EXAMPLE:

What is the horizontal distance $D$ covered by a baseball thrown at a
speed $v$ at an angle $\theta$ from the horizontal?

{\bf Step 1:} We are looking for a distance, $D$. The answer depends
on $v$ and $\theta$, the initial conditions of the baseball on its
release.  The answer also depends on the strength of gravity. Since
the baseball is thrown, we can assume that it doesn't go very high or
far, and we can ignore things like the variation of gravity with height
and the curvature of the Earth. So the answer also depends on $g$, the
assumed constant acceleration due to gravity. The complete set of
independent variables is $v,\theta,g$, and we are looking for
$D(v,\theta,g)$.

{\bf Step 2:} Distance is measured in {\it m}, speed in {\it m/s},
$\theta$ in {\it radians} which are dimensionless, and acceleration in
$m/s^2$. To make meters from the latter three quantities, simply divide
$v^2$ by $g$. So $Q = v^2/g$.


{\bf Step 3:} The only dimensionless quantity that can be formed from
the variables $v,\theta,g$ is $\theta$ itself.


{\bf Step 4:} The full solution to this problem must take the form $D=
v^2/g f(\theta)$. The dimensional analysis is done, but does the
result make sense? Check limits. If $v$ is increased, the baseball
goes farther, and if $g$ is increased it goes less far. This seems
reasonable.  Convince yourself that $f(\theta)\rightarrow0$ if either
$\theta\rightarrow 0$ or $\theta\rightarrow\pi/2$.

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