Because the "d" (pool radius) drops out of the equation, the child's pool can never be "small enough" to meet the parallel gravitational forces requirement. I get it now. 1.4 hour day for a perfectly flat southern pole pool, no matter what size the pool is. Very cool! Thanks. What shape would the Earth itself look like with a 1.4 hour day? Could it even hold together? Supposedly, the Earth originally had a 5 hour day pre-Moon. https://en.wikipedia.org/wiki/Giant_impact_hypothesis At 08:45 AM 3/6/2015, Eugene Salamin via math-fun wrote:

Within a few millimeters of the wall there is surface tension which usually pulls the liquid up (water on glass), but could pull it down (mercury on glass).

But then, this has nothing to do with Earth's rotation, and would make a tiny liquid mirror.

The gravitational acceleration g = 9.8 m s^-2 acting on a molecule on the surface at distance d from Earth's axis has a component gd/R pulling it inward, where R = 6.4e6 m is Earth's radius.

The centrifugal acceleration pulling it outward is ω^2 d where ω is the angular velocity.

In order to flatten the convex surface, these forces must be equal.

This requires ω = √(g/R) = 0.00124 s^-1.

The rotation period P = 2π/ω = 5080 s = 1.4 hours.

You need to spin faster than this to have a concave surface.

-- Gene

From: Henry Baker <hbaker1@pipeline.com> To: math-fun <math-fun@mailman.xmission.com> Cc: meekerdb <meekerdb@verizon.net> Sent: Friday, March 6, 2015 5:21 AM Subject: Re: [math-fun] newton rotating liquid mirror

No, I wouldn't expect a lake at the south pole to have a concave area in the center, but I would expect a child's round swimming pool at the south pole to have a concave surface.

There will be forces against the side of the pool to keep the edge of the water higher than the middle.

In the middle of a lake, these forces will be dissipated because the water will move away from the center.

At 09:40 PM 3/5/2015, meekerdb wrote: Would you expect a large lake at the south pole to have a concave area in the center, but match the geoid elsewhere?