GFD LabIV: Construction of parabolic turntable
If our cylindrical tank is filled with water, set turning and left until it comes in to solid body rotation, then the freesurface of the water will not be flat  it will be depressed in the middle and rise up slightly to its highest point along the rim of the tank  see
fig. below.
The shape of the free surface is given by
h(r) = h(0) + ((Ωērē)/(2g))
where h is the local depth, r is the distance from the axis or rotation, and
h(0) is the depth at r = 0. Thus the free surface takes on a parabolic shape.
Let's put in some numbers for our tank. We can obtain rotation rates of up to 10 rpm (which is a
Ω of 1 /s). The radius of the tank is
0.30 m and g = 9.81 m/s**2, giving
((Ωērē)/(2g)) ~ 5mm,
a small fraction of the depth to which the tank is typically filled.
It is very instructive to make the surface of our turntable parabolic. This can readily be achieved by filling a large flatbottomed pan with resin on a turntable and letting the resin harden while the turntable is left running
(10 rpm works well) for several hours  see here. The resulting parabolic surface can then be polished to create a low friction surface.
Place a ballbearing on the rotating parabolic surface  make sure that the table is rotating at the same speed as was used to create the parabola! Note that it does not fall in to the center, but instead finds a state of rest in which the component of gravitational
force resolved along the parabolic surface is exactly balanced by the outwarddirected horizontal component of the centrifugal
force.
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