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Re: curvature of buckets of water



Hi all-
Robert Cohen thoughtfully asks:
*********************************************************
On Mon, 5 Jul 1999, Leigh Palmer wrote:

Yes, there is a way to tell that the bucket is off-center.

The shape of the surface will be a paraboloid of revoution wherever
the bucket is placed on the turntable. The vertex of the paraboloid
will be on the axis of revolution. If that axis does not pass
through the bucket then the surface is called an off-axis paraboloid.

Why is it possible to tell if the bucket is off-center but it isn't
possible to tell (if one is stuck inside a box) where one is relative to
the axis of rotation (without more information given)?
*******************************************
This is a nice question because it has an analogue, that need
not concern us here, in general relativity.
I'll rephrase the question as follows: Is there a local measurement
that I can make at a point on a parabolic mirror to determine my distance
from the axis.
The answer is, yes. Here is the hint. Consider a parabola drawn on
a plane. Can I, by measuring the slope at a point on the parabola determine
my distance from the axis, if I know the direction of the axis? If the answer
is <no> (the slope is not zero), is there a further measurement at the point
that will give me the answer? Also, if I'm in a rotating frame of reference,
can I make local measurments that will determine the direction of the axis?
Regards,
Jack


"I scored the next great triumph for science myself,
to wit, how the milk gets into the cow. Both of us
had marveled over that mystery a long time. We had
followed the cows around for years - that is, in the
daytime - but had never caught them drinking fluid of
that color."
Mark Twain, Extract from Eve's
Autobiography