Bucket argument

# Bucket argument

Overview
Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

's rotating bucket argument (also known as "Newton's bucket") was designed 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 argument
Argument
In philosophy and logic, an argument is an attempt to persuade someone of something, or give evidence or reasons for accepting a particular conclusion.Argument may also refer to:-Mathematics and computer science:...

s 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
Operational definition
An operational definition defines something in terms of the specific process or set of validation tests used to determine its presence and quantity. That is, one defines something in terms of the operations that count as measuring it. The term was coined by Percy Williams Bridgman and is a part of...

of what is meant by "absolute rotation", and do not pretend to address the question of "rotation relative to what?".
Discussion

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Encyclopedia
Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

's rotating bucket argument (also known as "Newton's bucket") was designed 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 argument
Argument
In philosophy and logic, an argument is an attempt to persuade someone of something, or give evidence or reasons for accepting a particular conclusion.Argument may also refer to:-Mathematics and computer science:...

s 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
Operational definition
An operational definition defines something in terms of the specific process or set of validation tests used to determine its presence and quantity. That is, one defines something in terms of the operations that count as measuring it. The term was coined by Percy Williams Bridgman and is a part of...

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
Scholium
Scholia , are grammatical, critical, or explanatory comments, either original or extracted from pre-existing commentaries, which are inserted on the margin of the manuscript of an ancient author, as glosses. One who writes scholia is a scholiast...

at the very beginning of his great work, The Mathematical Principles of Natural Philosophy (1687), which established the foundations of classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

and introduced his law of universal gravitation, which yielded the first quantitatively adequate dynamical explanation of planetary motion. See the Principia on line at Andrew Motte Translation pp. 77–82.

Despite their embrace of the principle of rectilinear inertia
Inertia
Inertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...

and the recognition of the kinematical relativity of apparent motion (which underlies whether the Ptolemaic
Geocentric model
In astronomy, the geocentric model , is the superseded theory that the Earth is the center of the universe, and that all other objects orbit around it. This geocentric model served as the predominant cosmological system in many ancient civilizations such as ancient Greece...

or the Copernican
Heliocentrism
Heliocentrism, or heliocentricism, is the astronomical model in which the Earth and planets revolve around a stationary Sun at the center of the universe. The word comes from the Greek . Historically, heliocentrism was opposed to geocentrism, which placed the Earth at the center...

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
René Descartes
René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...

, and was supported (in part) by Gottfried Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

. 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.

Here is an everyday experience of the basic nature of the Descartes experiment: Consider sitting in your train and noticing a train originally at rest beside you in the railway station pulling away. Initially you think it is your own train accelerating, but then notice with surprise that you feel no force. Thus, it is not your own train moving, but the neighboring train. On the other hand, you would confirm your own train is accelerating if you sensed g-forces from the acceleration of your 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 Ernst Mach
Ernst Mach
Ernst Mach was an Austrian physicist and philosopher, noted for his contributions to physics such as the Mach number and the study of shock waves...

, who contended that all motion was relative.

## The argument

Newton discusses a bucket
Bucket
A bucket, also called a pail, is typically a watertight, vertical cylinder or truncated cone, with an open top and a flat bottom, usually attached to a semicircular carrying handle called the bail. A pail can have an open top or can have a lid....

filled with water
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...

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, 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? Newton says: "One can find out and measure the true and absolute circular motion of the water".

In the 1846 Andrew Motte translation of Newton's words:
The argument that the motion is absolute, not relative, is incomplete, as it limits the participants relevant to the experiment to only the pail and the water, a limitation that 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 arguing that only relative motion is established:
All observers agree that the surface of rotating water is curved. However, the explanation of this curvature involves centrifugal force for all observers with the exception of a truly stationary observer, who finds the curvature is consistent with the rate of rotation of the water as they observe it, with no need for an additional centrifugal force. Thus, a stationary frame can be identified, and it is not necessary to ask "Stationary with respect to what?":
A supplementary thought experiment
Thought experiment
A thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...

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
Rotating spheres
Isaac Newton's rotating spheres argument attempts to demonstrate that true rotational motion can be defined by observing the tension in the string joining two identical 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. 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. 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 Fg, the horizontal, radially outward centrifugal force FCfgl, and 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 stress
Shear stress
A shear stress, denoted \tau\, , is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section...

es. 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 Ω is the constant rate of rotation of the water. The gravitational force is unchanged at
where g is the acceleration due to gravity
Gravitational acceleration
In physics, gravitational acceleration is the acceleration on an object caused by gravity. Neglecting friction such as air resistance, all small bodies accelerate in a gravitational field at the same rate relative to the center of mass....

. These two forces add to make a resultant at an angle φ 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
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

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
Conservative force
A conservative force is a force with the property that the work done in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the net work done by a conservative force is zero.It is possible to define a numerical value of...

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
Mechanical work
In physics, work is a scalar quantity that can be described as the product of a force times the distance through which it acts, and it is called the work of the force. Only the component of a force in the direction of the movement of its point of application does 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
Mechanical equilibrium
A standard definition of static equilibrium is:This is a strict definition, and often the term "static equilibrium" is used in a more relaxed manner interchangeably with "mechanical equilibrium", as defined next....

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
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

form:

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

The principle of operation of the centrifuge
Centrifuge
A centrifuge is a piece of equipment, generally driven by an electric motor , that puts an object in rotation around a fixed axis, applying a force perpendicular to the axis...

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.

• Centrifugal force (rotating reference frame)
• Inertial frame of reference
Inertial frame of reference
In physics, an inertial frame of reference is a frame of reference that describes time homogeneously and space homogeneously, isotropically, and in a time-independent manner.All inertial frames are in a state of constant, rectilinear motion with respect to one another; they are not...

• Mach's principle
Mach's principle
In theoretical physics, particularly in discussions of gravitation theories, Mach's principle is the name given by Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach....

• Mechanics of planar particle motion
Mechanics of planar particle motion
This article describes a particle in planar motion when observed from non-inertial reference frames. The most famous examples of planar motion are related to the motion of two spheres that are gravitationally attracted to one another, and the generalization of this problem to planetary motion....

• Philosophy of space and time: Absolutism vs. relationalism
• Rotating reference frame
Rotating reference frame
A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth. A rotating frame of reference is a special case of a non-inertial reference...

• Rotating spheres
Rotating spheres
Isaac Newton's rotating spheres argument attempts to demonstrate that true rotational motion can be defined by observing the tension in the string joining two identical spheres...

• Sagnac effect
Sagnac effect
The Sagnac effect , named after French physicist Georges Sagnac, is a phenomenon encountered in interferometry that is elicited by rotation. The Sagnac effect manifests itself in a setup called ring interferometry. A beam of light is split and the two beams are made to follow a trajectory in...