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.^{ [1] } 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*?"^{ [2] } General relativity dispenses with absolute space and with physics whose cause is external to the system, with the concept of geodesics of spacetime.^{ [3] }

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:^{ [4] }^{ [5] }

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:^{ [6] }

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.^{ [2] }^{ [7] }

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.

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*.^{ [2] }

This sphere example was used by Newton himself to discuss the detection of rotation relative to absolute space.^{ [8] } 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.^{ [9] } (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. (See reactive centrifugal force.) The description of this system next is presented from the viewpoint of an inertial frame and from a rotating frame of reference.

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.

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.)^{ [10] } 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.^{ [11] }

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:^{ [12] }^{ [13] }

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:^{ [12] }

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}| = ω*R*.

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.

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:

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:

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:

or,

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).

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):^{ [14] }

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:

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

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:

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

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 ).^{ [15] } Combining the terms:^{ [16] }

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.

The isotropy of the cosmic background radiation is another indicator that the universe does not rotate.^{ [17] }

- ↑ See Louis N. Hand; Janet D. Finch (1998).
*Analytical Mechanics*. Cambridge University Press. p. 324. ISBN 0-521-57572-9. and I. Bernard Cohen; George Edwin Smith (2002).*The Cambridge companion to Newton*. Cambridge University Press. p. 43. ISBN 0-521-65696-6. - 1 2 3 Robert Disalle (2002). I. Bernard Cohen; George E. Smith (eds.).
*The Cambridge Companion to Newton*. Cambridge University Press. p. 43. ISBN 0-521-65696-6. - ↑ Gilson, James G. (September 1, 2004),
*Mach's Principle II*, arXiv: physics/0409010 , Bibcode:2004physics...9010G - ↑ See the
*Principia*on line at "Definitions".*The Principia*. Retrieved 2010-05-13. - ↑ Max Born (1962).
*Einstein's Theory of Relativity*. Courier Dover Publications. p. 80. ISBN 0-486-60769-0.inertial forces.

- ↑ Max Born (1962).
*Einstein's Theory of Relativity*(Greatly revised and enlarged ed.). Courier Dover Publications. p. 82. ISBN 0-486-60769-0.inertial forces.

- ↑ Ignazio Ciufolini; John Archibald Wheeler (1995).
*Gravitation and Inertia*. Princeton University Press. pp. 386–387. ISBN 0-691-03323-4. - ↑ Max Born (1962).
*Einstein's Theory of Relativity*. Courier Dover Publications. p. Figure 43, p. 79. ISBN 0-486-60769-0.inertial forces.

- ↑ D. Lynden-Bell (1996). Igorʹ Dmitrievich Novikov; Bernard Jean Trefor Jones; Draza Marković (eds.).
*Relativistic Astrophysics*. Cambridge University Press. p. 167. ISBN 0-521-62113-5. - ↑ Barry Dainton (2001).
*Time and Space*. McGill-Queen's Press. p. 175. ISBN 0-7735-2306-5. - ↑ Jens M. Knudsen & Poul G. Hjorth (2000).
*Elements of Newtonian Mechanics*. Springer. p. 161. ISBN 3-540-67652-X. - 1 2 Georg Joos & Ira M. Freeman (1986).
*Theoretical Physics*. New York: Courier Dover Publications. p. 233. ISBN 0-486-65227-0. - ↑ John Robert Taylor (2004).
*Classical Mechanics*. Sausalito CA: University Science Books. pp. 348–349. ISBN 1-891389-22-X. - ↑ Many sources are cited in Fictitious force. Here are two more: PF Srivastava (2007).
*Mechanics*. New Delhi: New Age International Publishers. p. 43. ISBN 978-81-224-1905-4. and NC Rana & PS Joag (2004).*Mechanics*. New Delhi: Tata McGraw-Hill. p. 99ff. ISBN 0-07-460315-9. - ↑ The case ω
_{S}< 0 applies to the earlier example with spheres at rest in the inertial frame. - ↑ This result can be compared with Eq. (3.3) in Stommel and Moore. They obtain the equation where and in their notation, and the left-hand side is the radial acceleration in polar coordinates according to the rotating observers. In this example, their Eq. (3.4) for the azimuthal acceleration is zero because the radius is fixed and there is no angular acceleration. See Henry Stommel; Dennis W. Moore (1989).
*An Introduction to the Coriolis Force*. Columbia University Press. p. 55. ISBN 0-231-06636-8.coriolis Stommel.

- ↑ R. B. Partridge (1995).
*3 K: The Cosmic Microwave Background Radiation*. Cambridge University Press. pp. 279–280. ISBN 0-521-35254-1., D. Lynden-Bell (1996).*Relativistic Astrophysics*(Igorʹ Dmitrievich Novikov, Bernard Jean Trefor Jones, Draza Marković (Editors) ed.). p. 167. ISBN 0-521-62113-5., and Ralph A. Alpher and Robert Herman (1975).*Big bang cosmology and the cosmic black-body radiation*(in*Proc. Am. Phil. Soc.*vol. 119, no. 5 (1975) ed.). pp. 325–348. ISBN 9781422371077.Henning Genz (2001).*Nothingness*. Da Capo Press. p. 275. ISBN 0-7382-0610-5.

In mechanics, **acceleration** is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities. The orientation of an object's acceleration is given by the orientation of the *net* force acting on that object. The magnitude of an object's acceleration, as described by Newton's Second Law, is the combined effect of two causes:

Isaac Newton's rotating **bucket argument** 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.

A **centripetal force** is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". In Newtonian mechanics, gravity provides the centripetal force causing astronomical orbits.

In physics, the **Coriolis force** is an inertial or fictitious force that acts on objects that are in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the left of the motion of the object. In one with anticlockwise rotation, the force acts to the right. Deflection of an object due to the Coriolis force is called the **Coriolis effect**. Though recognized previously by others, the mathematical expression for the Coriolis force appeared in an 1835 paper by French scientist Gaspard-Gustave de Coriolis, in connection with the theory of water wheels. Early in the 20th century, the term *Coriolis force* began to be used in connection with meteorology.

In classical physics and special relativity, an **inertial frame of reference** is a frame of reference that is not undergoing acceleration. In an inertial frame of reference, a physical object with zero net force acting on it moves with a constant velocity —or, equivalently, it is a frame of reference in which Newton's first law of motion holds. An inertial frame of reference can be defined in analytical terms as a frame of reference that describes time and space homogeneously, isotropically, and in a time-independent manner. Conceptually, the physics of a system in an inertial frame have no causes external to the system. An inertial frame of reference may also be called an **inertial reference frame**, **inertial frame**, **Galilean reference frame**, or **inertial space**.

In physics, a **frame of reference** consists of an abstract coordinate system and the set of physical reference points that uniquely fix the coordinate system and standardize measurements within that frame.

The **Rossby number** (**Ro**) named for Carl-Gustav Arvid Rossby, is a dimensionless number used in describing fluid flow. The Rossby number is the ratio of inertial force to Coriolis force, terms and in the Navier–Stokes equations respectively. It is commonly used in geophysical phenomena in the oceans and atmosphere, where it characterizes the importance of Coriolis accelerations arising from planetary rotation. It is also known as the **Kibel number**.

In physics, **circular motion** is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body.

In classical mechanics, **Euler's rotation equations** are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with its axes fixed to the body and parallel to the body's principal axes of inertia. Their general form is:

A **fictitious force** is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as an accelerating or rotating reference frame. An example is seen in a passenger vehicle that is accelerating in the forward direction – passengers perceive that they are acted upon by a force in the rearward direction pushing them back into their seats. An example in a rotating reference frame is the force that appears to push objects outwards towards the rim of a centrifuge. These apparent forces are examples of fictitious forces.

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.

In theoretical physics a **Coriolis field** is one of the *apparent* gravitational fields felt by a rotating or forcibly-accelerated body, together with the **centrifugal field** and the **Euler field**.

**Rotation around a fixed axis** is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible; if two rotations are forced at the same time, a new axis of rotation will appear.

The **Coriolis frequency***ƒ*, also called the **Coriolis parameter** or **Coriolis coefficient**, is equal to twice the rotation rate *Ω* of the Earth multiplied by the sine of the latitude *φ*.

**Rotational–vibrational coupling** occurs when the rotation frequency of an object is close to or identical to a natural internal vibration frequency. The animation on the right shows a simple example. The motion depicted in the animation is for the idealized situation that the force exerted by the spring increases linearly with the distance to the center of rotation. Also, the animation depicts what would occur if there would not be any friction.

**Inertial waves**, also known as **inertial oscillations**, are a type of mechanical wave possible in rotating fluids. Unlike surface gravity waves commonly seen at the beach or in the bathtub, inertial waves flow through the interior of the fluid, not at the surface. Like any other kind of wave, an inertial wave is caused by a restoring force and characterized by its wavelength and frequency. Because the restoring force for inertial waves is the Coriolis force, their wavelengths and frequencies are related in a peculiar way. Inertial waves are transverse. Most commonly they are observed in atmospheres, oceans, lakes, and laboratory experiments. Rossby waves, geostrophic currents, and geostrophic winds are examples of inertial waves. Inertial waves are also likely to exist in the molten core of the rotating Earth.

The **magnetorotational instability (MRI)** is a fluid instability that causes an accretion disk orbiting a massive central object to become turbulent. It arises when the angular velocity of a conducting fluid in a magnetic field decreases as the distance from the rotation center increases. It is also known as the **Velikhov–Chandrasekhar instability** or **Balbus–Hawley instability** in the literature, not to be confused with the electrothermal Velikhov instability. The MRI is of particular relevance in astrophysics where it is an important part of the dynamics in accretion disks.

In classical mechanics, the **Euler force** is the fictitious tangential force that appears when a non-uniformly rotating reference frame is used for analysis of motion and there is variation in the angular velocity of the reference frame's axes. The **Euler acceleration**, also known as **azimuthal acceleration** or **transverse acceleration** is that part of the absolute acceleration that is caused by the variation in the angular velocity of the reference frame.

In Newtonian mechanics, the **centrifugal force** is an inertial 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 parallel to the axis of rotation and passing through the coordinate system's origin. If the axis of rotation passes through the coordinate system's origin, the centrifugal force is directed radially outwards from that axis. The magnitude of centrifugal force *F* on an object of mass *m* at the distance *r* from the origin of a frame of reference rotating with angular velocity *ω* is:

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. See centrifugal force, two-body problem, orbit and Kepler's laws of planetary motion. Those problems fall in the general field of analytical dynamics, the determination of orbits from given laws of force. This article is focused more on the kinematical issues surrounding planar motion, that is, determination of the forces necessary to result in a certain trajectory *given* the particle trajectory. General results presented in fictitious forces here are applied to observations of a moving particle as seen from several specific non-inertial frames, for example, a *local* frame, and a *co-rotating* frame. The Lagrangian approach to fictitious forces is introduced.

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