Newton’s bucket is a cylindrical bucket that rotates at angular velocity . Due to viscosity, any liquid in the bucket is affected by the rotation, eventually achieving the exact same .
However, once in equilibrium, the liquid’s surface is not flat, but curved upwards from the center. This is due to the centrifugal force on a molecule with mass :
Where is the molecule’s position relative to the axis of rotation. This (fictitious) force can be written as the gradient of a potential , such that :
In addition, each molecule feels a gravitational force , where :
Overall, the molecule therefore feels an “effective” force with a potential given by:
At equilibrium, the hydrostatic pressure in the liquid is the one that satisfies:
Removing the gradients gives integration constants and , so the equilibrium equation is:
We isolate this for and rewrite , where is the liquid height at the center:
At the surface, we demand that , where is the air pressure. The -coordinate at which this is satisfied is as follows, telling us that the surface is parabolic:
- B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.