Piano Key Inertial Balance

[Mark Davidson]

My compliments to John for his very nice implementation.

John was considering measuring small parts also and attaching one of the
calibration weights to slow the whole thing down and make it easier to
measure.  I have been giving this some thought, and since it is probably of
(relatively) general interest, thought I'd comment here rather than in
private.

My general conclusion is that to measure small parts, you need a much weaker
spring and smaller calibration bars - on the same order of magnitude as the
parts you want to measure (i.e. 1000-2000 g cm^2).  Here's why.

Let's say you can measure the period accurately to .01 seconds.  Sounds
small, but when you count 20 periods and divide by 20 it's doable.  With my
setup, with a 20,000 g cm^2 bar attached, if I change the period by .01, I
get a change of approximately 300 g cm^2.  (Plug in the period you get for
this bar and the period + .01 and see what difference in MOI you get -
that's your resolution.) To measure a change of 0.1g in a hammer - the
resolution we've become accustomed to - requires measuring accurately to
about 1/2000 sec.

There are basically two ways to get around this problem.  The first is to be
able to measure time more accurately.  This is doable with electronic
sensors and timers (e.g. a photogate).  While a knowledgeable person could
probably build one cheaply, to purchase one outright is a few hundred
dollars.  There are very nice ones designed for measuring pendulums in
physics labs that could do the job very well.

The second way to get around the problem is to use a much weaker spring.  If
we look at the equation

I = T^2 / K - Ib

I = moment of inertia of part being measured
T = period
Ib = MOI of balance
K = constant

In this equation, K = 4 * pi^2 / kappa, where kappa is the torsional spring
constant.  We want K to be larger so that larger changes in T result in less
change in I, and the way to do this is to make kappa smaller (i.e. use
weaker spring).

Sounds like interchangeable springs might be a good thing to design in for
this purpose.  Some S-hooks on the strings where they attach to the springs
should make this easy enough.  Those wishing to use magnets might want to
keep this in mind also.

BTW, the implication is that you can also get better resolution for your
larger parts by using a weaker spring.

-Mark