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Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
's rotating bucket argument (also known as Newton's bucket) was designed to demonstrate that true
rotational motion Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
cannot be defined as the relative rotation of the body with respect to the immediately surrounding bodies. It is one of five
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialecti ...
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 Absolute may refer to: Companies * Absolute Entertainment, a video game publisher * Absolute Radio, (formerly Virgin Radio), independent national radio station in the UK * Absolute Software Corporation, specializes in security and data risk manage ...
. Alternatively, these experiments provide an
operational definition An operational definition specifies concrete, replicable procedures designed to represent a construct. In the words of American psychologist S.S. Stevens (1935), "An operation is the performance which we execute in order to make known a concept." F ...
of what is meant by "
absolute rotation In physics, the concept of absolute rotation—rotation independent of any external reference—is a topic of debate about relativity, cosmology, and the nature of physical laws. For the concept of absolute rotation to be scientifically meaningf ...
", and do not pretend to address the question of "rotation relative to ''what''?"
General relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
dispenses with absolute space and with physics whose cause is external to the system, with the concept of
geodesics In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
of
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
.


Background

These arguments, and a discussion of the distinctions between absolute and relative time, space, place and motion, appear in a scholium at the end of Definitions sections in Book I of Newton's work, '' The Mathematical Principles of Natural Philosophy'' (1687) (not to be confused with General Scholium at the end of Book III), which established the foundations of
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
and introduced his
law of universal gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distan ...
, which yielded the first quantitatively adequate dynamical explanation of
planetary motion In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a ...
. Despite their embrace of the principle of rectilinear
inertia Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
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 René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...
, and was supported (in part) by
Gottfried Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
. 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 ''t''1 found in the vicinity of one group of "landmark" bodies and at some ''t''2 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 were 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. For contingent reasons having to do with the
Inquisition The Inquisition was a group of institutions within the Catholic Church whose aim was to combat heresy, conducting trials of suspected heretics. Studies of the records have found that the overwhelming majority of sentences consisted of penances, ...
, Descartes spoke of motion as both absolute and relative. By the late 19th century, the contention that ''all motion is relative'' was re-introduced, notably by
Ernst Mach Ernst Waldfried Josef Wenzel Mach ( , ; 18 February 1838 – 19 February 1916) was a Moravian-born Austrian physicist and philosopher, who contributed to the physics of shock waves. The ratio of one's speed to that of sound is named the Mach n ...
(1883).


The argument

Newton discusses a
bucket A bucket is typically a watertight, vertical cylinder or truncated cone or square, with an open top and a flat bottom, attached to a semicircular carrying handle called the ''bail''. A bucket is usually an open-top container. In contrast, a ...
( la, situla) filled with
water Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as ...
hung by a cord.For a discussion of Newton's original argument, see 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:See the ''Principia'' on line a
Andrew Motte Translation
pp. 79-81
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: The degree in which Mach's hypothesis is integrated in general relativity is discussed in the article
Mach's principle In theoretical physics, particularly in discussions of gravitation theories, Mach's principle (or Mach's conjecture) is the name given by Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach. The hypothe ...
; it is generally held that general relativity is not entirely Machian. 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 A thought experiment is a hypothetical situation in which a hypothesis, theory, or principle is laid out for the purpose of thinking through its consequences. History The ancient Greek ''deiknymi'' (), or thought experiment, "was the most anc ...
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 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. The basis of the argument is that all observers make two observation ...
.


Detailed analysis

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 two 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, often denoted by ( Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. '' Normal stress'', on ...
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 A banked turn (or banking turn) is a turn or change of direction in which the vehicle banks or inclines, usually towards the inside of the turn. For a road or railroad this is usually due to the roadbed having a transverse down-slope towards the ...
, 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 In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors a ...
Fg + FCfgl.) As ''r'' increases, the centrifugal force increases according to the relation (the equations are written per unit mass): :F_ = m \Omega^2 r \ , where ''Ω'' is the constant rate of rotation of the water. The gravitational force is unchanged at :F_ = 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 :\tan \varphi =\frac = \frac \ , 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, :\tan \varphi = \frac \ , leading to the ordinary differential equation for the shape of the surface: :\frac = \frac \ , or, integrating: : h(r) =h(0) + \frac \left( \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.


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 In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potenti ...
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 Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization in ...
and has a potential energy of the form: :_ = -\frac 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: : F_ = -\frac _ = m \Omega^2 r \ . 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
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an animal t ...
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 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 h(r)\,: then the potential energy per unit mass contributed by gravity is g h(r) \ and the total potential energy per unit mass on the surface is : = _0 + gh(r) - \frac\Omega^2 r^2\, with _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: :h(r) = \fracr^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 A centrifuge is a device that uses centrifugal force to separate various components of a fluid. This is achieved by spinning the fluid at high speed within a container, thereby separating fluids of different densities (e.g. cream from milk) or ...
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.


See also

*
Centrifugal force In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
*
Inertial frame of reference In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
*
Mach's principle In theoretical physics, particularly in discussions of gravitation theories, Mach's principle (or Mach's conjecture) is the name given by Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach. The hypothe ...
*
Mechanics of planar particle motion This article describes a particle in planar motionSee for example, , when observed from non-inertial reference frames.''Fictitious forces'' (also known as a ''pseudo forces'', ''inertial forces'' or ''d'Alembert forces''), exist for observers i ...
* Philosophy of space and time: Absolutism vs. relationalism *
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. (This article considers only ...
*
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. The basis of the argument is that all observers make two observation ...
*
Rotational gravity Artificial gravity is the creation of an inertial force that mimics the effects of a gravitational force, usually by rotation. Artificial gravity, or rotational gravity, is thus the appearance of a centrifugal force in a rotating frame of ref ...
* Sagnac effect


References


Further reading

* * The isotropy of the
cosmic background radiation Cosmic background radiation is electromagnetic radiation from the Big Bang. The origin of this radiation depends on the region of the spectrum that is observed. One component is the cosmic microwave background. This component is redshifted ph ...
is another indicator that the universe does not rotate. See: ** ** **


External links


Newton's Views on Space, Time, and Motion
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.
Newton's Bucket
An interactive applet illustrating the water shape, and an attached PDF file with a mathematical derivation of a more complete water-shape model than is given in this article. {{Isaac Newton Classical mechanics Isaac Newton Thought experiments in physics