Re: curvature of buckets of water

[Leigh Palmer <palmer@SFU.CA>]

> On Mon, 5 Jul 1999, Leigh Palmer wrote:
>
> > Yes. The figure of the surface of an off-center bucket is simply an
> > off-axis paraboloid of revolution, a somewhat uncommonly employed
> > optical element. The segments of the Keck telescopes ao Mauna Kea
> > are all off-axis paraboloids.
>
> I apologize for my dual questions.  Do you mean that "yes, the shape will
> still be the same" or "yes, there is a way to tell that the bucket is
> off-center"?

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.

The rotation of the planet would have to be very fast to produce a
concavity in the surface of a bucket of water located at either pole.
At such a speed a fluid planet (a good approximation even for Earth)
would fly apart. (This tendency is said by some to be a response to
centrifugal force.)

> > Given one more piece of information (the period of rotation of the
> > planet) one could use a Foucault pendulum and a stopwatch to
> > determine one's exact latitude. In principle the curvature of the
> > water surface in a bucket held relatively stationary on the surface
> > of the planet will take the (convex) form of the planet's constant
> > potential surface in the rotating frame*. This can be seen through
> > a telescope by observing Jupiter or (more dramatically) Saturn. The
> > detailed nature of the surface could depend strongly on local
> > gravitational anomalies, of course, but the effect will be too
> > small to detect in a bucket. (See my earlier physics question.)
>
> How does one determine the period of rotation?  Also, the Foucault
> pendulum would just determine the orientation of local "vertical" relative
> to the rotation axis.  How does one know that the axis of rotation goes
> through the center of the planet?

I said that one would have to be given more information, and I don't
know any good method for determining the period of rotation while
inside a closed room. The period of rotation of the plane of
oscillation of a pendulum depends upon latitude as well as rotation
period, and it is a minimum at the poles.

Leigh