Re: Effect of Moon on balance -- electronic balances

["John S. Denker" <jsd@MONMOUTH.COM>]

There's less disagreement here than meets
the eye:

 -- It is 100% possible to get an electronic
balance that measures mass to high precision,
essentially as Michael E. described.
 -- It is 100% possible to get an electronic
scale that measures weight, essentially as
John C. described.
 -- It is also possible to get instruments
that measure some messy combination of mass
and weight.

And you can't necessarily tell by looking at
the instrument exactly what you've got.
Under ordinary terrestrial conditions the
distinctions don't matter much.  People
aren't usually very careful with the
terminology;  just today I used the word
"balance" when I should have used "scale".

M.E. makes an interesting point:
> The equal-arm balance compares two masses as the same time, but at
> slightly different locations.  The electronic balance compares two
> masses at the same location, but at slightly different times.

Let's skip the electronic-balance issue and
focus on the plain old "equal" arm balance.
(Actually the arms don't have to be equal.)
Such things have been in use for something
like 7000 years.  (Compare this with the
wheel, which has been in use for "only" 5500
years.)

As M.E. points out, real balances have some
nonidealities.  For one thing, their sensitivity
is proportional to |g|.  On top of that, if
the arm-length is deltaX, then there is a
systematic error proportional to
  (d/dx g) dot deltaX

Now the moon causes a gravity gradient, so
we see that even an arm-balance is perturbed
by the moon.  The effect is small, but it
exists in principle.

Now suppose you are an expert physicist and
you are hired to make a mass-meter that is
much more nearly ideal.  What would you do???

One good idea would be to make a symmetric
"flapping arms" balance, with two arms, one
to the left and one to the right, constrained
so that both arms move up together when the
measured mass moves down.  This would be
insensitive to (d/dx g).  To leading order
it would be sensitive to
   (d/dx_i d/dx_j g) deltaX_i deltaX_j
[where I have switched to component notation
because the more-abstract dot-product notation
is a crock for second-rank (and higher)
objects.] This nonideality will be smaller
than the previous nonideality by a factor of
something like deltaX/R, where R is the distance
to the moon.

Now let's raise the stakes.  Suppose your name
was John Harrison and you wanted to make a
chronometer that was
 -- insensitive to |g|,
 -- insensitive to temporal variations of g
    (due to straight-line jostling),
 -- insensitive to spatial variations of g
    (due to the centrifugal field resulting
    from yaw, pitch, and roll), and
 -- insensitive to variations in temperature.

Aboard ship the perturbation
 (d/dx_i g)
due to yaw/pitch/roll is enormous compared to
the perturbation due to the moon's tide-producing
field.  So you need a tremendously high degree of
insensitivity.

Harrison managed to do it.  The result
  http://www.harrisonclocks.co.uk/chronometers2.htm
is a masterpiece of pre-industrial-revolution
high technology.  Just amazing.

He did in fact use counter-rotating rotors.
He invented caged roller bearings.
He invented the bimetallic strip for temperature
compensation.
etc. etc. etc.
Pretty good considering he was trained as a
woodworker, and used wooden gears in his first
clocks.  Just imagine high-precision clocks
with wooden gears.

For more information, you can read
 _the illustrated Longitude_
by Sobel.  It's a wonderful story well told
(although I wish it went into more of the
technical details).

===========

Returning to the main thread:  Suppose you were
in a situation where the tide-producing force
was a million times bigger than what you're
used to.  Imagine being aboard a heaving ship,
or imagine the moon's orbit being a lot closer.
How would you measure mass accurately under such
conditions?  It might make for an interesting
class discussion, or perhaps a term-paper topic.