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Shape of the surface of a liquid being spun in a bowl

Determine the shape assumed by the surface of a liquid being spun in a circular bowl at constant angular velocity, W.

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If there is a net force acting on some point of a system then, by Newton's second law, you must have an acceleration there. This means that if a system is in static equilibrium, then at every point the sum of all forces must be zero. A convenient way to find the static equilibrium configuration is by demanding that the total potential energy of the system is minimal. If it is minimal, then a local infinitesimal perturbation somewhere will yield no change in the potential energy to first order in the perturbation. But since the change is also equal to the local perturbation times the net force acting on the point where the perturbation is applied, it follows that the net force there is zero.

The water isn't stationary in our case, but we can switch to co-rotating coordinates in which the bowl is stationary. In this rotating frame the laws of classical mechanics are valid as if it were a nonrotating frame, at the price of introducing fictitious forces. There is a Coriolis force which is velocity dependent and there is a centrifugal force which is directed away from the rotational axis. In this case we assume that a steady state has been reached in which the water is not moving relative to the co-rotating frame. So, we only have to deal with the ...

Solution Summary

A detailed explanation is given. We also explain the Lagrange multiplier method used in the derivation.