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Abstract

University physics courses teach that centrifugal force doesn't exist, while university applied maths courses teach that centrifugal force is merely a fictitious force that arises when making observations from a rotating frame of reference. Meanwhile, Sir Isaac Newton claimed that a centrifugal force is the equal and opposite reaction to a centripetal force. We also read in the literature that the centrifugal force acting on a body relative to a centre of rotation is merely an effect of inertia, owing to the tendency for the body to move in a straight-line path, and so it draws away from the centre. This article attempts to ascertain which, if any, of these positions is correct, and why the matter is important.
1
The Importance of Centrifugal Force
Frederick David Tombe,
Northern Ireland, United Kingdom,
sirius184@hotmail.com
28th May 2022
Abstract. University physics courses teach that centrifugal force doesn’t exist, while
university applied maths courses teach that centrifugal force is merely a fictitious force
that arises when making observations from a rotating frame of reference. Meanwhile,
Sir Isaac Newton claimed that a centrifugal force is the equal and opposite reaction to a
centripetal force.
We also read in the literature that the centrifugal force acting on a body relative to
a centre of rotation is merely an effect of inertia, owing to the tendency for the body to
move in a straight-line path, and so it draws away from the centre. This article attempts
to ascertain which, if any, of these positions is correct, and why the matter is important.
The Inertial Frame of Reference
I. It would be generally agreed that in an inertial frame of reference in a
terrestrial context, and in a non-rotating system, that centrifugal force is a
quantity which is not noticed to exist. In such a context, a body continues in its
state of rest or uniform straight-line motion unless acted upon by a Newtonian
force, and while we can geometrically identify a centrifugal force acting
outwards on a moving body, from every point in space, these centre-fleeing
centrifugal forces all cancel to zero. In a rotating system however, if a body is
forced to co-rotate with the system, then the specific centrifugal force from the
centre of rotation will become an isolated physical reality within the rotating
system, and an externally applied inward acting centripetal force will be
required to cancel it.
When physics students are taught that centrifugal force doesn’t exist, the
demonstration invariably takes place in the restricted context of circular motion
where the students are correctly shown how a centre-seeking centripetal force
deflects an object off its straight-line inertial path. Unfortunately, however, the
restricted context of circular motion masks the involvement of a centrifugal
force by virtue of the fact that this has the same magnitude as the centripetal
force, and this makes it superficially appear as though no centrifugal force is
involved. In order to expose the centrifugal force, we need to look at curved
path motion more generally, and we should ideally avoid circular motion
examples for the reasons stated in the previous sentence. Elliptical planetary
orbits serve this purpose very well.
2
The Gravitational Field
II. No physical explanation was given in the section above for an inertial frame
of reference, just as no physical explanation has ever been given for gravity. We
do however know that a gravitational field distorts an inertial frame of reference
and that the centrifugal force to the gravitational centre acts in opposition to
gravity. Both gravity and centrifugal force are radial forces in that they relate to
the second time derivative of the radial distance to a point origin. In the case of
gravity, this point origin is clearly defined by the location of the gravitational
mass of the attracting body, whereas in the case of centrifugal force, all points
in space are legitimate point origins. When doing orbital mechanics, we are of
course only interested in the centrifugal force to the gravitational centre or to
the focus of the orbit. These orbits are conic sections which can be ellipses,
circles, parabolas, or hyperbolas.
Gravity is accepted to be a real Newtonian force since it causes a body to
deviate from its state of rest or its uniform straight line inertial path. Centrifugal
force, on the other hand, is merely a consequence of the tendency of a moving
body to travel in a straight-line path in the absence of a centripetal force acting
on it. As such, centrifugal force is said to be an inertial force rather than a
Newtonian force.
Centrifugal force is nevertheless every bit as real as gravity. We have
known since the time of Leibniz that the radial planetary orbital equation takes
the form [1], [2], [3],
𝑚𝑟̈𝐫
̂= 𝑚(𝑔 + 𝑟θ
̇2)𝐫
̂ (1)
where 𝑚𝑔𝐫
̂ is gravity and 𝑚𝑟θ
̇2𝐫
̂ is centrifugal force. Equation (1) can be
rearranged as,
𝑚𝐠 = 𝑚(𝑟̈ − 𝑟θ
̇2)𝐫
̂ (2)
and due to conservation of angular momentum (Kepler’s second law),
equation (2) can be simplified to,
𝑚𝐠 = 𝑚𝐫̈ (3)
In modern applied maths textbooks, equation (3) is a popular starting point
in planetary orbital analysis, but it masks the involvement of centrifugal force
and leads to the false claim that gravity is the only real force acting in the
planetary system. The argument is that the 𝑚𝐫̈ term represents the total force in
the inertial frame of reference, and that this equates to gravity, and that therefore
no centrifugal force is involved. While it can be seen from equations (1) and (2)
3
that this argument is false, the argument is often reinforced by denying the name
‘centrifugal force’ to the 𝑚𝑟θ
̇2𝐫
̂ term. But it is centrifugal force, and it is very
much involved in a planetary orbit. Gravity and centrifugal force are both radial
forces acting in opposition to each other, and the latter is implicit within the
second time derivative of the radial position vector. Equation (3) is not an actual
equation for gravity, but rather it is telling us that gravity is the difference
between the net radial force and the centrifugal force.
Newton’s Rotating Bucket
III. Centrifugal force is physically encountered as a real force in Newton’s
rotating bucket. There is no doubt about its existence, and it is not, as Newton
claimed, a reaction to a centripetal force, [4], [5]. The centrifugal force acting
outwards against the inside of the bucket’s wall is the primary active agent,
while the inward acting centripetal force exerted by the wall is a reactive
constraint. The centrifugal force is quite definitely not a reaction. Centrifugal
force is an action, just as we already saw in the section above in connection with
planetary orbits where it is totally independent of gravity. Centrifugal force is
tied up with a body’s momentum and kinetic energy and it is as real as those.
Some say that centrifugal force is merely the inertial resistance to attempts
by a centripetal force to deviate a body off its straight-line inertial path. It’s true
that centripetal force curved path motion and it’s true that centrifugal force
opposes this centripetal force. However, the centripetal force does not actually
induce the centrifugal force, and in many cases, such as described in the
paragraph above, it’s the centripetal force that is induced by the centrifugal
force. The centrifugal force is there already, built into the momentum of the
moving body. It’s the body’s momentum that decides the magnitude of the
centrifugal force. This magnitude is not decided by the magnitude of any
applied or induced centripetal forces. Centrifugal force is not, as Newton said,
part of an action-reaction pair with the centripetal force, [6], and besides, the two
forces are acting on the same body, whereas an action-reaction pair can only act
across two bodies. When a centrifugal force forms part of an action-reaction
pair, it is with another centrifugal force.
Neither is centrifugal force merely a fictitious force arising due to making
observations from a rotating frame of refence. We don’t need a rotating frame
of reference to observe a real centrifugal force, although in the case of a body
that is being forced to co-rotate, an actual physically rotating system can
directly expose the centrifugal force relative to the centre of rotation. A
centrifuge device is a prime illustration of this syndrome, and we can observe a
centrifuge in action while sitting in a chair in the corner of the room. We don’t
need to be rotating with the centrifuge in order to observe the centrifugal force
in action.
4
Conclusion
IV. The importance of centrifugal force lies principally in the context of a
vortex, as illustrated by Newton’s rotating bucket. The outward pressure which
the rotating water exerts on the inside wall of the bucket is due to a very real
centrifugal force, which is in turn due to the inertial effect of each element of
the water as it endeavours to move in a straight-line path. The reason why this
context is so important is because it lies at the root of the principle upon which
James Clerk Maxwell established his famous equations of electromagnetism.
Maxwell’s equations were established on the basis that space is densely packed
with tiny molecular vortices that press against each other with centrifugal force
while striving to dilate, [7], [8], [9], [10], [11]. These vortices self-align along their
mutual rotation axes hence tracing out the local magnetic field.
When a charged particle moves through a magnetic field, it experiences a
magnetic force, F = qv×B, which is actually a centrifugal force, or more
accurately, the difference between two centrifugal forces acting oppositely to
each other on either side of the particle, at right angles to its direction of motion.
This sea of molecular vortices serves as the physical basis for an inertial frame
of reference, [12], as well as serving as the medium for the propagation of light
waves, and the magnetic flux density B is related to the fine-grained vorticity in
the sea.
References
[1] Tombe, F.D., “Leibniz’s Radial Planetary Orbital Equation” (2017)
https://www.researchgate.net/publication/320694417_Leibniz's_Radial_Planetary_Orbital_Equation
[2] Goldstein, Herbert, “Classical Mechanics” Equation 3-12, page 74, Addison Wesley Publishing
Company (1980)
https://books.google.co.uk/books?id=9M8QAQAAIAAJ&q=intitle:classical+intitle:mechanics+inauth
or:goldste
in+3-12&dq=intitle:classical+intitle:mechanics+inauthor:goldstein+3-12&hl=en
[3] Taylor, John R, “Classical Mechanics” Equation 8.37, page 306, University Science Books,
Colorado (2005)
https://books.google.co.uk/books?id=P1kCtNr-
pJsC&pg=PA306&lpg=PA306&dq=taylor+classical+mechanics+centrifugal+force&source=bl&ots=
giRy46Kvt
2&sig=9JBDTR4SDoL-wvgR9w-
Ug2vXjEQ&hl=en&sa=X&ved=0CEAQ6AEwBmoVChMIhpz955P5xgIVzG0UCh2HXQB-
#v=onepage&q=taylor%20classical%20mechanics%20centrifugal%20force&f=false
[4] Tombe, F.D., “The Reality of Centrifugal Force”, (2021)
https://www.researchgate.net/publication/350060937_The_Reality_of_Centrifugal_Force
[5] Tombe, F.D., Aether Friction in the Planetary Orbits” (2021)
https://www.researchgate.net/publication/350873624_Aether_Friction_in_the_Planetary_Orbits
5
[6] Swetz, Frank J., “Learn From The Masters” ‘An Episode in the History of Celestial Mechanics’,
page 269, Mathematical Association of America (1995)
https://books.google.co.uk/books?id=gqGLoh-
WYrEC&pg=PA269&dq=reaction+fictitious+rotating+frame+%22centrifugal+force%22&hl=en#v=o
nepage&q
=reaction%20fictitious%20rotating%20frame%20%22centrifugal%20force%22&f=false
[7] Clerk-Maxwell, J., “On Physical Lines of Force”, Equations (5) and (77), Philosophical
Magazine, Volume XXI, Fourth Series, London, (1861)
http://vacuum-physics.com/Maxwell/maxwell_oplf.pdf
[8] Tombe, F.D., “The Double Helix Theory of the Magnetic Field” (2006)
Galilean Electrodynamics, Volume 24, Number 2, p.34, (March/April 2013)
https://www.researchgate.net/publication/295010637_The_Double_Helix_Theory_of_the_Magnetic_
Field
[9] Whittaker, E.T., “A History of the Theories of Aether and Electricity”, Chapter 4, pages 100-102,
(1910)
“All space, according to the younger Bernoulli, is permeated by a fluid aether, containing an
immense number of excessively small whirlpools. The elasticity which the aether appears to possess,
and in virtue of which it is able to transmit vibrations, is really due to the presence of these
whirlpools; for, owing to centrifugal force, each whirlpool is continually striving to dilate, and so
presses against the neighbouring whirlpools.”
[10] O’Neill, John J., “PRODIGAL GENIUS, Biography of Nikola Tesla”, Long Island, New York,
15th July 1944, Fourth Part, paragraph 23, quoting Tesla from his 1907 paper “Man’s Greatest
Achievement” which was published in 1930 in the Milwaukee Sentinel,
“Long ago he (mankind) recognized that all perceptible matter comes from a primary substance, of a
tenuity beyond conception and filling all space - the Akasha or luminiferous ether - which is acted
upon by the life-giving Prana or creative force, calling into existence, in never ending cycles, all
things and phenomena. The primary substance, thrown into infinitesimal whirls of prodigious velocity,
becomes gross matter; the force subsiding, the motion ceases and matter disappears, reverting to the
primary substance”.
http://www.rastko.rs/istorija/tesla/oniell-tesla.html
http://www.ascension-research.org/tesla.html
[11] Lodge, Sir Oliver, “Ether (in physics)”, Encyclopaedia Britannica,
Fourteenth Edition, Volume 8, Pages 751-755, (1937)
http://gsjournal.net/Science-
Journals/Historical%20PapersMechanics%20/%20Electrodynamics/Download/4105
In relation to the speed of light, The most probable surmise or guess at present is that the ether is a
perfectly incompressible continuous fluid, in a state of fine-grained vortex motion, circulating with
that same enormous speed. For it has been partly, though as yet incompletely, shown that such a
vortex fluid would transmit waves of the same general nature as light waves i.e., periodic
disturbances across the line of propagationand would transmit them at a rate of the same order of
magnitude as the vortex or circulation speed”
[12] Tombe, F.D., “Straight Line Motion” (2018)
https://www.researchgate.net/publication/325472420_Straight_Line_Motion
... In simple terms, the reversal torque is due to the fact that an inertial force has been partially blocked by a Newtonian force, leaving the remainder to cause an inertial torque. This torque acts transversely out of the plane of rotation, [3]. ...
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The purpose of presenting this standard derivation, largely copied from applied maths notes taken at Queen's University, Belfast, in 1979, is to show that centrifugal force is as real as gravity, and that no rotating frame of reference is necessary in order to observe it.
Equation 3-12, page 74
  • Herbert Goldstein
Goldstein, Herbert, "Classical Mechanics" Equation 3-12, page 74, Addison Wesley Publishing Company (1980) https://books.google.co.uk/books?id=9M8QAQAAIAAJ&q=intitle:classical+intitle:mechanics+inauth or:goldste in+3-12&dq=intitle:classical+intitle:mechanics+inauthor:goldstein+3-12&hl=en
Classical Mechanics" Equation 8
  • John R Taylor
Taylor, John R, "Classical Mechanics" Equation 8.37, page 306, University Science Books, Colorado (2005) https://books.google.co.uk/books?id=P1kCtNr-