Bucket argument

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Bucket_argument

Isaac Newton'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 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?" General relativity dispenses with absolute space and with physics whose cause is external to the system, with the concept of geodesics of spacetime.

Background

These arguments, and a discussion of the distinctions between absolute and relative time, space, place and motion, appear in a General Scholium at the very beginning of Newton's 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.

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.

Detection of rotation: red flags pop out on flexible arms when either object actually rotates. A: Central object rotates. B: Outer ring rotates, but in opposite direction. C: Both rotate, but in opposite directions. D: Both are locked together and rotate in the same direction.

 

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, Descartes spoke of motion as both absolute and relative. However, his real position was that motion is absolute.

By the late 19th century, the contention that all motion is relative was re-introduced, notably by Ernst Mach (1883).

When, accordingly, we say that a body preserves unchanged its direction and velocity in space, our assertion is nothing more or less than an abbreviated reference to the entire universe.

— Ernst Mach; as quoted by Ciufolini and Wheeler: Gravitation and Inertia, p. 387

The argument

Newton discusses a bucket (Latin: situla) 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, 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:

If a vessel, hung by a long cord, is so often turned about that the cord is strongly twisted, then filled with water, and held at rest together with the water; after, by the sudden action of another force, it is whirled about in the contrary way, and while the cord is untwisting itself, the vessel continues for some time this motion; the surface of the water will at first be plain, as before the vessel began to move; but the vessel by gradually communicating its motion to the water, will make it begin sensibly to revolve, and recede by little and little, and ascend to the sides of the vessel, forming itself into a concave figure...This ascent of the water shows its endeavour to recede from the axis of its motion; and the true and absolute circular motion of the water, which is here directly contrary to the relative, discovers itself, and may be measured by this endeavour. ... And therefore, this endeavour does not depend upon any translation of the water in respect to ambient bodies, nor can true circular motion be defined by such translation. ...; but relative motions...are altogether destitute of any real effect. ...It is indeed a matter of great difficulty to discover, and effectually to distinguish, the true motions of particular bodies from the apparent; because the parts of that immovable space in which these motions are performed, do by no means come under the observations of our senses.

— Isaac Newton; Principia, Book 1: Scholium

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:

Newton's experiment with the rotating vessel of water simply informs us that the relative rotation of the water with respect to the sides of the vessel produces no noticeable centrifugal forces, but that such forces are produced by its relative rotations with respect to the mass of the earth and other celestial bodies.

— Ernst Mach, as quoted by L. Bouquiaux in Leibniz, p. 104

The degree in which Mach's hypothesis is integrated in general relativity is discussed in the article Mach's principle; 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?":

The original question, "relative to what frame of reference do the laws of motion hold?" is revealed to be wrongly posed. For the laws of motion essentially determine a class of reference frames, and (in principle) a procedure for constructing them.

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

The interface of two immiscible liquids rotating around a vertical axis is an upward-opening circular paraboloid.

 

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. 

Force diagram for an element of water surface in co-rotating frame. Top: Radial section and selected point on water surface; the water, the co-rotating frame, and the radial section share a constant angular rate of rotation given by the vector Ω. Bottom: Force diagram at selected point on surface. The slope of the surface adjusts to make all three forces sum to zero.

 

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 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. To quote Anthony and Brackett:

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 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 Ω, the fictitious centrifugal force is conservative and has a potential energy of the form:

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

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.

Rotating spheres

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/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 observations: the tension in the string joining the bodies (which is the same for all observers) and the rate of rotation of the spheres (which is different for observers with differing rates of rotation). Only for the truly non-rotating observer will the tension in the string be explained using only the observed rate of rotation. For all other observers a "correction" is required (a centrifugal force) that accounts for the tension calculated being different from the one expected using the observed rate of rotation. 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?" General relativity dispenses with absolute space and with physics whose cause is external to the system, with the concept of geodesics of spacetime.

Background

Newton was concerned to address the problem of how it is that we can experimentally determine the true motions of bodies in light of the fact that absolute space is not something that can be perceived. Such determination, he says, can be accomplished by observing the causes of motion (that is, forces) and not simply the apparent motions of bodies relative to one another (as in the bucket argument). As an example where causes can be observed, if two globes, floating in space, are connected by a cord, measuring the amount of tension in the cord, with no other clues to assess the situation, alone suffices to indicate how fast the two objects are revolving around the common center of mass. (This experiment involves observation of a force, the tension). Also, the sense of the rotation —whether it is in the clockwise or the counter-clockwise direction— can be discovered by applying forces to opposite faces of the globes and ascertaining whether this leads to an increase or a decrease in the tension of the cord (again involving a force). Alternatively, the sense of the rotation can be determined by measuring the apparent motion of the globes with respect to a background system of bodies that, according to the preceding methods, have been established already as not in a state of rotation, as an example from Newton's time, the fixed stars.

In the 1846 Andrew Motte translation of Newton's words:

We have some arguments to guide us, partly from the apparent motions, which are the differences of the true motions; partly from the forces, which are the causes and effects of the true motions. For instance, if two globes kept at a given distance one from the other, by means of a cord that connects them, were revolved about their common center of gravity; we might, from the tension of the cord, discover the endeavor of the globes to recede from the axis of their motion. ... And thus we might find both the quantity and the determination of this circular motion, even in an immense vacuum, where there was nothing external or sensible with which the globes could be compared.

— Isaac Newton, Principia, Book 1, Scholium

To summarize this proposal, here is a quote from Born:

If the earth were at rest, and if, instead, the whole stellar system were to rotate in the opposite sense once around the earth in twenty-four hours, then, according to Newton, the centrifugal forces [presently attributed to the earth's rotation] would not occur.

— Max Born: Einstein's Theory of Relativity, pp. 81-82

Mach took some issue with the argument, pointing out that the rotating sphere experiment could never be done in an empty universe, where possibly Newton's laws do not apply, so the experiment really only shows what happens when the spheres rotate in our universe, and therefore, for example, may indicate only rotation relative to the entire mass of the universe.

For me, only relative motions exist…When a body rotates relatively to the fixed stars, centrifugal forces are produced; when it rotates relatively to some different body and not relative to the fixed stars, no centrifugal forces are produced.

— Ernst Mach; as quoted by Ciufolini and Wheeler: Gravitation and Inertia, p. 387

An interpretation that avoids this conflict is to say that the rotating spheres experiment does not really define rotation relative to anything in particular (for example, absolute space or fixed stars); rather the experiment is an operational definition of what is meant by the motion called absolute rotation.

Figure 1: Two spheres tied with a string and rotating at an angular rate ω. Because of the rotation, the string tying the spheres together is under tension.

 

Figure 2: Exploded view of rotating spheres in an inertial frame of reference showing the centripetal forces on the spheres provided by the tension in the tying string.

 

Formulation of the argument

This sphere example was used by Newton himself to discuss the detection of rotation relative to absolute space. Checking the fictitious force needed to account for the tension in the string is one way for an observer to decide whether or not they are rotating – if the fictitious force is zero, they are not rotating. (Of course, in an extreme case like the gravitron amusement ride, you do not need much convincing that you are rotating, but standing on the Earth's surface, the matter is more subtle.) Below, the mathematical details behind this observation are presented. 

Figure 1 shows two identical spheres rotating about the center of the string joining them. The axis of rotation is shown as a vector Ω with direction given by the right-hand rule and magnitude equal to the rate of rotation: |Ω| = ω. The angular rate of rotation ω is assumed independent of time (uniform circular motion). Because of the rotation, the string is under tension. The description of this system next is presented from the viewpoint of an inertial frame and from a rotating frame of reference.

Inertial frame

Adopt an inertial frame centered at the midpoint of the string. The balls move in a circle about the origin of our coordinate system. Look first at one of the two balls. To travel in a circular path, which is not uniform motion with constant velocity, but circular motion at constant speed, requires a force to act on the ball so as to continuously change the direction of its velocity. This force is directed inward, along the direction of the string, and is called a centripetal force. The other ball has the same requirement, but being on the opposite end of the string, requires a centripetal force of the same size, but opposite in direction. See Figure 2. These two forces are provided by the string, putting the string under tension, also shown in Figure 2.

Rotating frame

Adopt a rotating frame at the midpoint of the string. Suppose the frame rotates at the same angular rate as the balls, so the balls appear stationary in this rotating frame. Because the balls are not moving, observers say they are at rest. If they now apply Newton's law of inertia, they would say no force acts on the balls, so the string should be relaxed. However, they clearly see the string is under tension. (For example, they could split the string and put a spring in its center, which would stretch.) To account for this tension, they propose that in their frame a centrifugal force acts on the two balls, pulling them apart. This force originates from nowhere – it is just a "fact of life" in this rotating world, and acts on everything they observe, not just these spheres. In resisting this ubiquitous centrifugal force, the string is placed under tension, accounting for their observation, despite the fact that the spheres are at rest.

Coriolis force

What if the spheres are not rotating in the inertial frame (string tension is zero)? Then string tension in the rotating frame also is zero. But how can that be? The spheres in the rotating frame now appear to be rotating, and should require an inward force to do that. According to the analysis of uniform circular motion:

${\displaystyle \mathbf {F} _{\mathrm {centripetal} }=-m\mathbf {\Omega \ \times } \left(\mathbf {\Omega \times x_{B}} \right)\ }$

${\displaystyle =-m\omega ^{2}R\ \mathbf {u} _{R}\ ,}$

where u_R is a unit vector pointing from the axis of rotation to one of the spheres, and Ω is a vector representing the angular rotation, with magnitude ω and direction normal to the plane of rotation given by the right-hand rule, m is the mass of the ball, and R is the distance from the axis of rotation to the spheres (the magnitude of the displacement vector, |x_B| = R, locating one or the other of the spheres). According to the rotating observer, shouldn't the tension in the string be twice as big as before (the tension from the centrifugal force plus the extra tension needed to provide the centripetal force of rotation)? The reason the rotating observer sees zero tension is because of yet another fictitious force in the rotating world, the Coriolis force, which depends on the velocity of a moving object. In this zero-tension case, according to the rotating observer the spheres now are moving, and the Coriolis force (which depends upon velocity) is activated. According to the article fictitious force, the Coriolis force is:

${\displaystyle \mathbf {F} _{\mathrm {fict} }=-2m{\boldsymbol {\Omega }}\times \mathbf {v} _{B}\ }$

${\displaystyle =-2m\omega \left(\omega R\right)\ \mathbf {u} _{R},}$

where R is the distance to the object from the center of rotation, and v_B is the velocity of the object subject to the Coriolis force, |v_B| = ωR. 

In the geometry of this example, this Coriolis force has twice the magnitude of the ubiquitous centrifugal force and is exactly opposite in direction. Therefore, it cancels out the ubiquitous centrifugal force found in the first example, and goes a step further to provide exactly the centripetal force demanded by uniform circular motion, so the rotating observer calculates there is no need for tension in the string − the Coriolis force looks after everything.

General case

What happens if the spheres rotate at one angular rate, say ω_I (I = inertial), and the frame rotates at a different rate ω_R (R = rotational)? The inertial observers see circular motion and the tension in the string exerts a centripetal inward force on the spheres of:

${\displaystyle \mathbf {T} =-m\omega _{I}^{2}R\mathbf {u} _{R}\ .}$

This force also is the force due to tension seen by the rotating observers. The rotating observers see the spheres in circular motion with angular rate ω_S = ω_I − ω_R (S = spheres). That is, if the frame rotates more slowly than the spheres, ω_S > 0 and the spheres advance counterclockwise around a circle, while for a more rapidly moving frame, ω_S < 0, and the spheres appear to retreat clockwise around a circle. In either case, the rotating observers see circular motion and require a net inward centripetal force:

${\displaystyle \mathbf {F} _{\mathrm {Centripetal} }=-m\omega _{S}^{2}R\mathbf {u} _{R}\ .}$

However, this force is not the tension in the string. So the rotational observers conclude that a force exists (which the inertial observers call a fictitious force) so that:

${\displaystyle \mathbf {F} _{\mathrm {Centripetal} }=\mathbf {T} +\mathbf {F} _{\mathrm {Fict} }\ ,}$

or,

${\displaystyle \mathbf {F} _{\mathrm {Fict} }=-m\left(\omega _{S}^{2}R-\omega _{I}^{2}R\right)\mathbf {u} _{R}\ .}$

The fictitious force changes sign depending upon which of ω_I and ω_S is greater. The reason for the sign change is that when ω_I > ω_S, the spheres actually are moving faster than the rotating observers measure, so they measure a tension in the string that actually is larger than they expect; hence, the fictitious force must increase the tension (point outward). When ω_I < ω_S, things are reversed so the fictitious force has to decrease the tension, and therefore has the opposite sign (points inward).

Is the fictitious force ad hoc?

The introduction of F_Fict allows the rotational observers and the inertial observers to agree on the tension in the string. However, we might ask: "Does this solution fit in with general experience with other situations, or is it simply a "cooked up" ad hoc solution?" That question is answered by seeing how this value for F_Fict squares with the general result (derived in Fictitious force):

${\displaystyle \mathbf {F} _{\mathrm {Fict} }=-2m{\boldsymbol {\Omega }}\times \mathbf {v} _{B}-m{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times \mathbf {x} _{B})}$ ${\displaystyle \ -m{\frac {d{\boldsymbol {\Omega }}}{dt}}\times \mathbf {x} _{B}\ .}$

The subscript B refers to quantities referred to the non-inertial coordinate system. Full notational details are in Fictitious force. For constant angular rate of rotation the last term is zero. To evaluate the other terms we need the position of one of the spheres:

${\displaystyle \mathbf {x} _{B}=R\mathbf {u} _{R}\ ,}$

and the velocity of this sphere as seen in the rotating frame:

${\displaystyle \mathbf {v} _{B}=\omega _{S}R\mathbf {u} _{\theta }\ ,}$

where u_θ is a unit vector perpendicular to u_R pointing in the direction of motion. 

The frame rotates at a rate ω_R, so the vector of rotation is Ω = ω_R u_z (u_z a unit vector in the z-direction), and Ω × u_R = ω_R (u_z × u_R) = ω_R u_θ ; Ω × u_θ = −ω_R u_R. The centrifugal force is then:

${\displaystyle \mathbf {F} _{\mathrm {Cfgl} }=-m{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times \mathbf {x} _{B})=m\omega _{R}^{2}R\mathbf {u} _{R}\ ,}$

which naturally depends only on the rate of rotation of the frame and is always outward. The Coriolis force is

${\displaystyle \mathbf {F} _{\mathrm {Cor} }=-2m{\boldsymbol {\Omega }}\times \mathbf {v} _{B}=2m\omega _{S}\omega _{R}R\mathbf {u} _{R}}$

and has the ability to change sign, being outward when the spheres move faster than the frame ( ω_S > 0 ) and being inward when the spheres move slower than the frame ( ω_S < 0 ). Combining the terms:

${\displaystyle \mathbf {F} _{\mathrm {Fict} }=\mathbf {F} _{\mathrm {Cfgl} }+\mathbf {F} _{\mathrm {Cor} }}$ 
${\displaystyle =\left(m\omega _{R}^{2}R+2m\omega _{S}\omega _{R}R\right)\mathbf {u} _{R}=m\omega _{R}\left(\omega _{R}+2\omega _{S}\right)R\mathbf {u} _{R}}$

${\displaystyle =m(\omega _{I}-\omega _{S})(\omega _{I}+\omega _{S})\ R\mathbf {u} _{R}=-m\left(\omega _{S}^{2}-\omega _{I}^{2}\right)\ R\mathbf {u} _{R}.}$

Consequently, the fictitious force found above for this problem of rotating spheres is consistent with the general result and is not an ad hoc solution just "cooked up" to bring about agreement for this single example. Moreover, it is the Coriolis force that makes it possible for the fictitious force to change sign depending upon which of ω_I, ω_S is the greater, inasmuch as the centrifugal force contribution always is outward.

Rotation and cosmic background radiation

The isotropy of the cosmic background radiation is another indicator that the universe does not rotate.