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A fluid in a rotating bucket will take on a parabolic shape (for example of some simple derivations of this result see http://en.wikipedia.org/wiki/Bucket_argument). The assumptions that play into the derivations that I've seen do not take into consideration a Coriolis effect. Suppose one did not know that the earth was spinning. Is it possible to measure the angular frequency of the earth using a bucket of fluid by considering how the Coriolis effect modified the curvature of the paraboloid in question?

Perhaps it's possible to optically measure the curvature of the paraboloid setup by the spinning process and from that determine the angular frequency of the planets rotation. I am concerned though that the effect of the rotation of the earth is so minuscule on the parabolic shape that it would be impossible to physically measure it.

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Note that even if the effect was measurable with acceptable noise, you still need to know your latitude. Think of the limiting case where the spinning mirror is right over one of the poles. There would be no coriolis effect (assuming convection and flow within the liquid itself have died down). –  Olin Lathrop Dec 25 '13 at 17:34

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This case is discussed in the following article by Paul Hickson:
Eliminating the coriolis effect in liquid mirrors

The Large Zenith Telescope is a 6 meter diameter Mercury mirror telescope. As part of the design process Hickson did a theoretical exporation of how the Coriolis effect would affect the mirror shape.

According to Hickson the Coriolis effect is significant for the LZT. Also, Hickson's calculations indicated the effect could be reduced below detection level by setting the telescope at a tilt of 13.1 arcsec.

For another Mercury mirror telescope, a 3 meter device in operation at the time the article was writtten (2001), Hickson calculated a required tilt of 12 arcsec. But "the effects were smaller than the atmospheric seeing and could not be adequately assessed". For the LZT: "The larger mirror diameter and better image sampling should allow us to verify the technique using this telescope".

I don't know whether this was followed up.

I gather from this that the accuracy of the paraboloid can only be assessed indirectly, from it's effects on the quality of the imaging.

Of course, it would be very difficult to obtain a value for the Earth rotation rate in this way. I gather that even for the 6 meter mirror the effect is close to being swamped by atmospheric seeing. The error in the angle of tilt that yields the best imaging will be something like 5 or 10 arcsec.

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Minor comment to the post (v3): In the future please link to abstract pages rather than pdf files, e.g., arxiv.org/abs/astro-ph/0108306 –  Qmechanic Dec 25 '13 at 16:43
@Qmechanic I assume the advantage of linking to the abstract page is that it serves as a permalink. If a new version of the article becomes available the abstract page will offer a link to the latest version. –  Cleonis Dec 25 '13 at 16:56
Great find! Thank you. –  ulmo Dec 25 '13 at 20:43
I've posted a followup question. physics.stackexchange.com/q/91315/17198 I find I disagree with Paul Hickson on a specific point. My take is that while the rotation of Earth effect on the Mercury mirror can be mitigated with a tilt of the axis, it cannot be eliminated. –  Cleonis Dec 25 '13 at 22:08

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