Editing Bucket argument (section)

===Potential energy===
The shape of the water's surface can be found in a different, very intuitive way using the interesting idea of the [[potential energy]] associated with the centrifugal force in the co-rotating frame.
In a reference frame uniformly rotating at angular rate Ω, the fictitious centrifugal force is [[conservative force|conservative]] and has a potential energy of the form:<ref name=Carmichael>{{cite book |page=[https://archive.org/details/theoryrelativit00carmgoog/page/n84 78] |author=Robert Daniel Carmichael |url=https://archive.org/details/theoryrelativit00carmgoog |quote=fictitious Christoffel potential. |title=The Theory of Relativity |publisher=John Wiley & Sons |year=1920}}</ref><ref name=Weber>{{cite book |title=Essential mathematical methods for physicists |author=Hans J. Weber & George B. Arfken |isbn=0-12-059877-9 |year=2003 |publisher=Academic Press |page=79 |url=https://books.google.com/books?id=k046p9v-ZCgC&dq=reverse+sign+centrifugal&pg=PA79}}</ref>
:<math>{U}_{\mathrm{Cfgl}} = -\frac{1}{2} m \Omega^2 r^2 \ ,</math>

where ''r'' is the radius from the axis of rotation. This result can be verified by taking the gradient of the potential to obtain the radially outward force:
:<math> F_{\mathrm{Cfgl}} = -\frac{\partial}{\partial r} {U}_{\mathrm{Cfgl}} </math> <math>= m \Omega^2 r \ . </math>
The meaning of the potential energy (stored work) is that movement of a test body from a larger radius to a smaller radius involves doing [[Mechanical work|work]] against the centrifugal force and thus gaining potential energy. But this test body at the smaller radius where its elevation is lower has now lost equivalent gravitational potential energy.

Potential energy therefore explains the concavity of the water surface in a rotating bucket. Notice that at [[Mechanical equilibrium|equilibrium]] the surface adopts a shape such that an element of volume at any location on its surface has the same potential energy as at any other. That being so, no element of water on the surface has any incentive to move position, because all positions are equivalent in energy. That is, equilibrium is attained. On the other hand, were surface regions with lower energy available, the water occupying surface locations of higher potential energy would move to occupy these positions of lower energy, inasmuch as there is no barrier to lateral movement in an ideal liquid.

We might imagine deliberately upsetting this equilibrium situation by somehow momentarily altering the surface shape of the water to make it different from an equal-energy surface. This change in shape would not be stable, and the water would not stay in our artificially contrived shape, but engage in a transient exploration of many shapes until non-ideal frictional forces introduced by sloshing, either against the sides of the bucket or by the non-ideal nature of the liquid, killed the oscillations and the water settled down to the equilibrium shape.

To see the principle of an equal-energy surface at work, imagine gradually increasing the rate of rotation of the bucket from zero. The water surface is flat at first, and clearly a surface of equal potential energy because all points on the surface are at the same height in the gravitational field acting upon the water. At some small angular rate of rotation, however, an element of surface water can achieve lower potential energy by moving outward under the influence of the centrifugal force; think of an object moving with the force of gravity closer to the Earth's center: the object lowers its potential energy by complying with a force. Because water is incompressible and must remain within the confines of the bucket, this outward movement increases the depth of water at the larger radius, increasing the height of the surface at larger radius, and lowering it at smaller radius. The surface of the water becomes slightly concave, with the consequence that the potential energy of the water at the greater radius is increased by the work done against gravity to achieve the greater height. As the height of water increases, movement toward the periphery becomes no longer advantageous, because the reduction in potential energy from working with the centrifugal force is balanced against the increase in energy working against gravity. Thus, at a given angular rate of rotation, a concave surface represents the stable situation, and the more rapid the rotation, the more concave this surface. If rotation is arrested, the energy stored in fashioning the concave surface must be dissipated, for example through friction, before an equilibrium flat surface is restored.

To implement a surface of constant potential energy quantitatively, let the height of the water be <math>h(r)\,</math>: then the potential energy per unit mass contributed by gravity is <math>g h(r) \ </math> and the total potential energy per unit mass on the surface is
:<math>{U} = {U}_0 + gh(r) - \frac{1}{2}\Omega^2 r^2\,</math>

with <math>{U}_0</math> the background energy level independent of ''r''. In a static situation (no motion of the fluid in the rotating frame), this energy is constant independent of position ''r''.  Requiring the energy to be constant, we obtain the [[Parabola|parabolic]] form:
:<math>h(r) = \frac{\Omega^2}{2g}r^2 + h(0) \ , </math>

where ''h''(0) is the height at ''r'' = 0 (the axis). See Figures 1 and 2.

The principle of operation of the [[centrifuge]] also can be simply understood in terms of this expression for the potential energy, which shows that it is favorable energetically when the volume far from the axis of rotation is occupied by the heavier substance.