Is it possible to use the parabolic shape of a rotating fluid to measure the angular frequency of the rotation of the Earth? 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.  
 A: 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.
