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A closed vessel full of water is rotating with constant angular velocity $\Omega$ about a horizontal axis. Show that the surfaces of equal pressure are circular cylinders whose common axis is at a height $g/{\Omega}^2$ above the axis of rotation. Any ideas? I do not know how to start. |
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Use the assumption that water rotates at the same angular velocity as the vessel. Consider a small cube in an arbitrary point of the vessel. Consider the forces acting on the cube: the force of gravity and the force caused by pressure. As the cube is small, the force caused by pressure can be expressed via spatial derivatives of the pressure. Together the forces make the cube move with centripetal acceleration, so you can find the spatial derivatives of pressure $\frac{\partial P}{\partial x}$ and $\frac{\partial P}{\partial y}$ from Newton's second law. |
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