Dimensional analysis is a very powerful way of reasoning about problems
that can give order of magnitude estimates for quantities of
interest. For example, if we wanted to get a rough estimate of the
mass of the spherical drop of water in example 2.3.3 then
we can get it to within a factor of ten as follows.
That's good enough for us at present. We've decided to
wave our hands around a lot. Why is such an answer useful? Often
you're not even that sure of the size of the droplet and
you just want to know if something is about 
or 
.
First we want to know what are the important variables in the problem.
There's the density 
, which has units 
(mass per length cubed),
and also r radius, which has units of L (length). We want to
figure out how the mass is related to these two quantities. We guess
that the mass should equal some constant times 
. p
and q are unknown numbers that we'll try to determine. If we
are successful in determining them, then we'll almost have the answer.
You might be
wondering why we made this guess. It'll become clearer as we work through
this example.
Unfortunately there's that pesky constant that we won't be able to get, but most
of the time such constant don't differ from unity by more than a
factor of 10.
OK so how do we determine p and q? We can do so by keeping track of units. We write
Here we've introduced the proportionality symbol `` 
''.
This is useful for situations of this kind when we don't want
to be bothered by constants. The right hand side has units of
and the left hand side has units of M.
We better have the same units on the left and right hand side,
which then says that
The left hand side 
. The only way of getting the
left and right hand sides to match is to have p = 1 and
. Solving this gives q = 3.
So now we have p and q and so plugging this into
eq. 2.10 gives 
.
Note that 2.5 is of the same form, but there
we calculated the constant to be 
. So if we just
ignore the constant, that is set it equal to one, then
we end up off by roughly a factor 4, which gives the
right answer to within an order of magnitude.
