Re: curvature of buckets of water

[Robert A Cohen <bbq@ESU.EDU>]

My comments are interspersed below...

On Mon, 5 Jul 1999, Gary Karshner wrote:

> > I have a question about "absolute rotation."  If one placed the bucket of
> > water in the center of the rotating platform, one gets a curved surface.
> > If one places the bucket of water off-center of the rotating platform, one
> > also gets a curved surface but the orientation of the surface will be
> > different.  Will the shape still be the same?  That is, if one is in the
> > bucket's frame of reference, is there any way one can tell that the bucket
> > is off-center rather than just "tipped over" (away from what appears to be
> > the center of rotation)?
>
> What do you mean by the same shape? It will still be a parabola, but the
> focus will be at the center of rotation, i.e. the lowest water level in the
> bucket will be as close to the axis as it can get.

How does one tell where the lowest water level is without some reference
as to what is "up"?  After all, from the bucket's point of view, if one
defines the vertical as the orientation of a plumb bob, the plumb bob
would indicate that the *bucket* is tipped rather than the parabola.

> > In a related matter, suppose one is trapped inside a box on a foreign
> > planet. Is there an experiment that person can do to determine whether
> > they are located coincident with the axis of rotation (i.e., north or
> > south pole) rather than at some other latitude?
>
> Here I think a Foucault pendulum would suffice, if it rotates 360 degrees
> then you are at the pole but if it librates back and forth your not there.

Wouldn't the pendulum rotate 360 degrees no matter where you are, just
taking longer at some points than others?  Even at the equator, where the
pendulum doesn't rotate at all, how does one know that the planet isn't
rotating about the horizontal?

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"?

> 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?

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| Robert Cohen               Department of Physics       |
|                            East Stroudsburg University |
| bbq@esu.edu                East Stroudsburg, PA  18301 |
| http://www.esu.edu/~bbq/   (570) 422-3428              |
|                            **note new area code**      |
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