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Bucket argument
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Isaac Newton's rotating bucket argument (also known as "Newton's bucket") attempts to demonstrate that true rotational motion cannot be defined as the relative rotation of the body with respect to the immediately surrounding bodies. It is one of five arguments from the "properties, causes, and effects" of true motion and rest that support his contention that, in general, true motion and rest cannot be defined as special instances of motion or rest relative to other bodies, but instead can be defined only by reference to absolute space.
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Isaac Newton's rotating bucket argument (also known as "Newton's bucket") attempts to demonstrate that true rotational motion cannot be defined as the relative rotation of the body with respect to the immediately surrounding bodies. It is one of five arguments from the "properties, causes, and effects" of true motion and rest that support his contention that, in general, true motion and rest cannot be defined as special instances of motion or rest relative to other bodies, but instead can be defined only by reference to absolute space. Alternatively, these experiments provide an operational definition of what is meant by "absolute rotation", and do not pretend to address the question of "rotation relative to what?".
Background
These arguments, and a discussion of the distinctions between absolute and relative time, space, place and motion, appear in a Scholium at the very beginning of his great work, The Mathematical Principles of Natural Philosophy (1687), which established the foundations of classical mechanics and introduced his law of universal gravitation, which yielded the first quantitatively adequate dynamical explanation of planetary motion. See the Principia on line at pp. 77-82.
Despite their embrace of the principle of rectilinear inertia and the recognition of the kinematical relativity of apparent motion (which underlies whether the Ptolemaic or the Copernican system is correct), natural philosophers of the seventeenth century continued to consider true motion and rest as physically separate descriptors of an individual body. The dominant view Newton opposed was devised by René Descartes, and was supported (in part) by Gottfried Leibniz. It held that empty space is a metaphysical impossibility because space is nothing other than the extension of matter, or, in other words, that when one speaks of the space between things one is actually making reference to the relationship that exists between those things and not to some entity that stands between them. Concordant with the above understanding, any assertion about the motion of a body boils down to a description over time in which the body under consideration is at t1 found in the vicinity of one group of "landmark" bodies and at some t2 is found in the vicinity of some other "landmark" body or bodies.
Descartes recognized that there would be a real difference, however, between a situation in which a body with movable parts and originally at rest with respect to a surrounding ring was itself accelerated to a certain angular velocity with respect to the ring, and another situation in which the surrounding ring was given a contrary acceleration with respect to the central object. With sole regard to the central object and the surrounding ring, the motions would be indistinguishable from each other assuming that both the central object and the surrounding ring were absolutely rigid objects. However, if neither the central object nor the surrounding ring were absolutely rigid then the parts of one or both of them would tend to fly out from the axis of rotation. People who have noticed a train originally at rest beside them in the railway station pulling away from them, and have soon thereafter noticed with surprise that it is not their train that remains parked at the station, have experienced the basic nature of the Descartes experiment. Frequently these observers first question their initial impressions when they sense g forces from the acceleration of their own train.
For contingent reasons having to do with the Inquisition, Descartes spoke of motion as both absolute and relative. However, his real position was that motion is absolute.
A contrasting position was taken by Mach, who contended that all motion was relative.
The argument
Newton discusses a bucket filled with water hung by a cord. If the cord is twisted up tightly on itself and then the bucket is released, it begins to spin rapidly, not only with respect to the experimenter, but also in relation to the water it contains. (This situation would correspond to diagram B above.)
Although the relative motion at this stage is the greatest, the surface of the water remains flat, indicating that the parts of the water have no tendency to recede from the axis of relative motion, despite proximity to the pail. Eventually, as the cord continues to unwind, the surface of the water assumes a concave shape as it acquires the motion of the bucket spinning relative to the experimenter. This concave shape shows that the water is rotating, that despite the fact that the water is at rest relative to the pail. In other words, it is not the relative motion of the pail and water that causes concavity of the water, contrary to the idea that motions can only be relative, and that there is no absolute motion. (This situation would correspond to diagram D.) Possibly the concavity of the water shows rotation relative to something else: say absolute space? The argument is incomplete, as it limits the participants relevant to the experiment to only the pail and the water, which has not been established. In fact, the concavity of the water clearly involves gravitational attraction, and by implication the Earth also is a participant. Here is a critique due to Mach:
In the 1846 Andrew Motte translation of Newton's words:
A supplementary thought experiment with the same objective of determining the occurrence of absolute rotation also was proposed by Newton: the example of observing two identical spheres in rotation about their center of gravity and tied together by a string. Occurrence of tension in the string is indicative of absolute rotation; see Rotating spheres.
Detailed analysis
Of course, the historic interest of the rotating bucket experiment is its usefulness in suggesting one can detect absolute rotation by observation of the shape of the surface of the water. See Figure 1. However, one might question just how rotation brings about this change. Below are three approaches to understanding the concavity of the surface of rotating water in a bucket.
Newton's laws of motion
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. Figure 2 sets up the analysis in the co-rotating frame where the water appears stationary. The height of the water h is a function of the radial distance r from the axis of rotation O, h = h(r), and the aim is to determine this function. An element of water volume on the surface is shown to be subject to three forces: (i) The vertical force due to gravity Fg, (ii) The horizontal, radially outward centrifugal force FCfgl and (iii) The force normal to the surface of the water Fn 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. To quote Anthony and Brackett: 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 Fg + FCfgl.)
As r increases, the centrifugal force increases according to the relation (the equations are written per unit mass):
where O is the constant rate of rotation of the water. The gravitational force is unchanged at
with g = acceleration due to gravity. These two forces add to make a resultant at an angle f from the vertical given by
which clearly becomes larger as r increases. To insure 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,
leading to the ordinary differential equation for the shape of the surface:
or, integrating:
where h(0) is the height of the water at r = 0. In words, the surface of the water is parabolic in its dependence upon the radius.
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 O, the fictitious centrifugal force is conservative and has a potential energy of the form:
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:
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 : then the potential energy per unit mass contributed by gravity is and the total potential energy per unit mass on the surface is
with 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:
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.
Fluid mechanics
A more complete discussion of the rotating bucket is based upon fluid mechanics. For example, what is the explanation of the concave water surface in an inertial frame of reference? In this frame, the water is observed to rotate, and therefore executes uniform circular motion. This motion requires a centripetal force, and this centripetal force is supplied for a surface element of water by the horizontal component of the normal force from the water beneath. (The vertical component counteracts gravity.) This force from the water in turn derives from the walls of the bucket, which force the adjacent water to move in a circle by supplying centripetal force. That force is transmitted by the incompressible water to its surface.
This explanation is consistent with what has been said so far, but here is an unanswered question: at the vertical and bottom walls of the bucket, the walls transmit a constraining force to the adjacent water that causes that water to move in circular motion; but how does the shape of the bucket affect the solution?
The answer to this question comes from the equation of motion of a volume element in the fluid. Consider an elementary volume of fluid of mass per unit volume ? with a velocity v. The motion of this element of fluid is described by Euler's equation:
where p is the pressure inside the element of fluid and g is the acceleration due to gravity.
For the rotating fluid in the bucket rotating at angular rate O, assume for the moment that water rotates like a rigid body. Then the velocity of an elementary volume of water is given by:
with a unit vector perpendicular to the radial direction pointing in the direction of rotation and a unit vector in the radial direction. Euler's equation becomes:
where gravity is taken to be in the downward (negative) vertical z-direction with unit vector . To find a curve of constant pressure, suppose ds is a displacement along such a curve. In order that no change in pressure occur, we require the pressure gradient to be orthogonal to the displacement:
or,
Thus, a curve of constant pressure is given by:
which resembles the earlier result for the free concave surface of water. The pressure at a point (r, z) is:
and is the same for all angles ?. The constant of integration c is found by locating the free surface. At the free surface, the pressure is zero, and z = h(r):
as before, with the depth of water at r = 0 determining c as
This solution uses an assumed form for the velocity of the water that is only approximate. (The water is assumed to rotate like a rigid body.) As a result, the solution does not account for the behavior near the bottom of the bucket or the transition from horizontal to vertical walls supposing, say, the bucket has the shape of a half-cylinder. The approximate solution actually does not apply all the way to the bucket walls, but is valid only in the central region of the bucket, as described next.
Lamb points out that the assumed velocity of the liquid has a non-zero curl, and hence cannot be realized in an ideal liquid that cannot support tangential forces. Another way to put this is that the velocity cannot be represented as the gradient of a velocity potential, it is not irrotational, a rather anomalous situation. In particular, it can be shown that any motion generated from rest by impulsive pressure only is necessarily irrotational. For example, if we begin with a stationary fluid with a flat surface, its velocity has zero curl. Then it can be shown that any subsequent velocity configuration obtained by gradually increasing the rotation of the water also will have zero curl. To escape this predicament, one approach is to assume the angular velocity of the liquid is not O but some angular velocity ? = ?(r) that varies with radial position. In this case a velocity with zero curl can be "cooked up". These Rankine vortex or Lamb-Oseen vortex solutions splice together a concave region near the center of the bucket resembling the solution just found, and a convex solution outside this core region extending to the bucket wall. Once the existence of a velocity potential is established, for an incompressible fluid this potential satisfies Laplace's equation, opening the way to application of all its methods of solution and incorporation of whatever boundary conditions the bucket may impose. In this way solutions may be obtained that reflect the shape of the bucket, and do not depart significantly from the above results for the free surface in the central region of the bucket assuming the water has sufficient depth and the bucket has sufficient radius.
For a real bucket and a real liquid, the solution near the bucket walls may become quite complicated, involving boundary layers and a realistic model for the liquid. The Navier-Stokes equations must be solved, not just Laplace's equation.
Further reading
The isotropy of the cosmic background radiation is another indicator that the universe does not rotate. See , , and
External links
- from Stanford Encyclopedia of Philosophy, article by Robert Rynasiewicz. At the end of this article, loss of fine distinctions in the translations as compared to the original Latin text is discussed.
- see section on
Space, Time and Indiscernibles for Leibniz arguing against the idea of space acting as a causal agent.
See also
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