Bucket argument

The shape of the surface of a rotating liquid in a bucket can be determined using Newton's laws for the various forces on an element of the surface. For example, see Knudsen and Hjorth.[16] The analysis begins with the free body diagram in the co-rotating frame where the water appears stationary. The height of the water h = h(r) is a function of the radial distance r from the axis of rotation Ω, and the aim is to determine this function. An element of water volume on the surface is shown to be subject to three forces: the vertical force due to gravity F_g, the horizontal, radially outward centrifugal force F_Cfgl, and the force normal to the surface of the water F_n due to the rest of the water surrounding the selected element of surface. The force due to surrounding water is known to be normal to the surface of the water because a liquid in equilibrium cannot support shear stresses.[17] To quote Anthony and Brackett:[18]

The surface of a fluid of uniform density..., if at rest, is everywhere perpendicular to the lines of force; for if this were not so, the force at a point on the surface could be resolved into two components, one perpendicular and the other tangent to the surface. But from the nature of a fluid, the tangential force would set up a motion of the fluid, which is contrary to the statement that the fluid is at rest.

— William Arnold Anthony & Cyrus Fogg Brackett: Elementary Text-book of Physics, p. 127

Moreover, because the element of water does not move, the sum of all three forces must be zero. To sum to zero, the force of the water must point oppositely to the sum of the centrifugal and gravity forces, which means the surface of the water must adjust so its normal points in this direction. (A very similar problem is the design of a banked turn, where the slope of the turn is set so a car will not slide off the road. The analogy in the case of rotating bucket is that the element of water surface will "slide" up or down the surface unless the normal to the surface aligns with the vector resultant formed by the vector addition F_g + F_Cfgl.)

As r increases, the centrifugal force increases according to the relation (the equations are written per unit mass):

${\displaystyle F_{\mathrm {Cfgl} }=m{\mathit {\Omega }}^{2}r\ ,}$

where Ω is the constant rate of rotation of the water. The gravitational force is unchanged at

${\displaystyle F_{\mathrm {g} }=mg\ ,}$

where g is the acceleration due to gravity. These two forces add to make a resultant at an angle φ from the vertical given by

${\displaystyle \tan \varphi ={\frac {F_{\mathrm {Cfgl} }}{F_{\mathrm {g} }}}={\frac {{\mathit {\Omega }}^{2}r}{g}}\ ,}$

which clearly becomes larger as r increases. To ensure that this resultant is normal to the surface of the water, and therefore can be effectively nulled by the force of the water beneath, the normal to the surface must have the same angle, that is,

${\displaystyle \tan \varphi ={\frac {\mathrm {d} h}{\mathrm {d} r}}\ ,}$

leading to the ordinary differential equation for the shape of the surface:

${\displaystyle {\frac {\mathrm {d} h}{\mathrm {d} r}}={\frac {{\mathit {\Omega }}^{2}r}{g}}\ ,}$

or, integrating:

${\displaystyle h(r)=h(0)+{\frac {1}{2g}}\left({\mathit {\Omega }}r\right)^{2}\ ,}$

where h(0) is the height of the water at r = 0. In other words, the surface of the water is parabolic in its dependence upon the radius.

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 and has a potential energy of the form:[19][20]

${\displaystyle {U}_{\mathrm {Cfgl} }=-{\frac {1}{2}}m\Omega ^{2}r^{2}\ ,}$

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:

${\displaystyle F_{\mathrm {Cfgl} }=-{\frac {\partial }{\partial r}}{U}_{\mathrm {Cfgl} }}$ ${\displaystyle =m\Omega ^{2}r\ .}$

The meaning of the potential energy is that movement of a test body from a larger radius to a smaller radius involves doing work against the centrifugal force.

The potential energy is useful, for example, in understanding the concavity of the water surface in a rotating bucket. Notice that at 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. 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 ${\displaystyle h(r)\,}$: then the potential energy per unit mass contributed by gravity is ${\displaystyle gh(r)\ }$ and the total potential energy per unit mass on the surface is

${\displaystyle {U}={U}_{0}+gh(r)-{\frac {1}{2}}\Omega ^{2}r^{2}\,}$

with ${\displaystyle {U}_{0}}$ 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 parabolic form:

${\displaystyle h(r)={\frac {\Omega ^{2}}{2g}}r^{2}+h(0)\ ,}$

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.