Recently there was some discussion of how to teach
about gears and gear ratios. This prompted me to dust
off my spreadsheet that analyzes the gear-ratios of a
bike.
I will mainly talk about the 21-speed version. The
other version is for your convenience, if you want
to analyze a different bike ... and also to show
that what I say about the 21-speed bike is not a
fluke associated with a particular make or model.
My analysis starts from the hypothesis that as you
shift through the gears, it would be "nice" if each
shift produced about the same amount of change in
the gear ratio.
This leads to an obvious question as to how we should
quantify the "amount" of gear ratio. I suggest that
a logarithmic scale makes sense. In particular, the
dB unit (decibel) has a convenient size.
(Specifically, I am using units of 20*log10(ratio),
even though 10*log10(ratio) would arguably be more
properly called a dB, but I'm not sufficiently
motivated to change or even rename my units.)
Sure enough, when you do the analysis, you find that
to a remarkably good approximation, each step in the
rear gear cluster multiplies the gear ratio by a factor
of 1.15, i.e. it adds 1.21 dB. If you plot the overall
log(ratio) versus step-number, you get a remarkably
straight line.
Continuing the analysis, we find to a decent approximation
that each step on the front chainring has the same effect
as two steps on the rear cluster.
That means that this bike, which is nominally a 21-speed
bike, really has only 11 speeds (separated by ten steps).
The overall spread in gear ratio is very nearly a factor
of four. We can model the entire behavior by saying
each step corresponds to the tenth root of 4. All 21
shifter settings fall very close to what this model
predicts.
There's one slight twist. We have a lovely model ...
but the data does not quite fit the model as accurately
as it could. For starters, the steps on the front
chainring would be more uniform if it were configured
not as 26-36-46 but rather as 26-35-46 ... or yet more
uniform if it were 26-35-47.
The philosophy-of-physics lesson here is that just
because you have a model doesn't mean the data is
obliged to conform to the model. All too often students
seem to live in some dream-world where theories are
always right, and if the data doesn't fit the model
the data must be wrong. (Hah! Guess again!)
But let's get real. The laws of nature and of man
do not require bicycles to have evenly-spaced gears.
You might argue that 26-36-46 is "close enough", which
of course it is ... but most things on this bike are way
more precise than they really need to be, so I must say
I'm surprised that the chainring is not 26-35-46 or -47;
it just doesn't match the finesse and sophistication of
most things you find on a nice bike. Perhaps there is
some constraint I'm not aware of. Or perhaps it's just
an oversight; maybe they never plotted things out the
way I did.
You can use the spreadsheet to compare the as-sold scheme
to what might have been, by setting the "mode" cell to
1 or 0 respectively.
Another strange lesson has to do with how remarkably
well you can do approximating a transcendental function
(in this case the logarithm) using integers. The shifter
moves in integral steps, and there are an integral number
of teeth on each gear, yet the agreement between the data
and the tenth-root-of-four model is tighter than you might
have expected.
Another philosophy-of-physics lesson is that even an
imperfect model is better than nothing. When I'm teaching
a kid to ride a mountain bike, I tell him each step on
the front chainring is equivalent to two steps on the rear
cluster. It's not exaaaactly true, but it's close enough.
And just the idea that the gears "should" be evenly spaced
on a semi-log graph gives us a framework for presenting and
discussing the data. Even if there are exceptions and
deviations, we at least have a framework for discussing them.
To repeat: Don't just take a bunch of data and file it
away in a logbook somewhere. Find some way to analyze it.
Find some way to visualize it. Check whether it follows
the expected pattern or not. Analyze the data before (!)
you take the measuring apparatus apart, because the
analysis may well motivate you to repeat or extend the
measurements.
=================
As a tangent, talking about the tenth root of 4 reminds
me of the twelfth root of 2, and well-tempered music
scales. Some of the aforementioned lessons can be
revisited and reinforced:
-- It is remarkable how close you can come to rational
numbers such as a perfect fifth (3/2) and a perfect
fourth (4/3) by taking integer powers of the twelfth
root of two. (The major third doesn't do so well.)
-- Then, having done that, you find that the data for a
real piano doesn't fit the theory (because the strings
are anharmonic, due primarily to the stiffness of the steel).
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