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Re: curvature of buckets of water



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)?

A planet doesn't spin nearly fast enough to produce a detectable
curvature. Gravity and centrifugal force dominate over any effect
due to centrifugal force gradient *per se*, and (if I've thought
this out properly) it is the *rate of change* of the centrifugal
force gradient with respect to distance which is responsible for
that aspect of the curvature (its deviation from sphericity)
which one would have to detect to determine the distance from the
axis. That's two derivatives removed from detectability.

In summary:

Water in a bucket in a closed room on the surface of a planet
will have (in the lowest approximation) a plane surface oriented
perpendicularly to the vector sum of the gravitational and
centrifugal forces.

At distances from the poles which are large compared to the size
of the bucket there will be a very small curvature of this
surface which will be approximately ellipsoidal, with two
principal radii of curvature. Information about the direction to
the pole can be determined (in principle if not in practice)
from these parameters.

At smaller distances from the poles the principal radii differ
by smaller amounts, and they are equal at the poles. The surface
is approximately spherical there. It corresponds to the surface
of water in a bucket spun at a rate of one revolution per day.

To measure distance to the pole in this way one would have to go
one degree of accuracy further and measure the departure from
simple radii of curvature for the surface.

Too much!

Leigh