The Problem of the Spinning Tube
It’s been a while since I posted a problem in the folder for cute physics problems so here’s a nice little one for you to have a go at:
A vertical cylindrical tube of height 12cm and radius 6cm, sealed at the bottom and open at the top, is two-thirds filled with a liquid and set rotating with a constant angular velocity ω about a vertical axis. Neglecting the surface tension of the liquid, estimate the greatest angular velocity for which the liquid does not spill over the edge of the tube.
Answers through the comments box please!
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In the Dark 
November 22, 2017 at 8:34 pm
Nice problem indeed. The answer seems to be 2*sqrt(g*H*(1-alpha))/R ~= 21 sec^{-1} where alpha is the fraction of the cylinder and H,R are the height and radius of the cylinder.
November 22, 2017 at 10:30 pm
Yes, the angular frequency is (2/R)[g(H-D)]^1/2, where H is the cylinder height, D is the static depth of water, R is the cylinder radius, and g is the acceleration due to gravity.
November 23, 2017 at 3:29 pm
Yes. The original version of this question was divided into two parts, the first of which asked to that when spinning the surface deformed into a paraboloid of revolution (i.e. of the form z=cr^2), before asking the question above. I thought that made it too easy so cut it out!