Transcript The Strange Tale of Centrifugal Force

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The Strange Tale of
Centrifugal Force
History of Science Society
Annual Meeting
Austin, Texas
November 2004
William M. Shields
Science and Technology Studies
Virginia Tech
[email protected]
What is Centrifugal Force?
In classical mechanics, CF is the force
experienced by a body in an accelerated
reference frame characterized by a central
attractive force.
Magnitude proportional to the square of the
tangential velocity and inversely proportional
to the radius of the orbit, direction is radial
away from the center of the orbit.
Fc = mvt2/r
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Frames of Reference
In the inertial frame of reference used by
Newton, only centripetal force acts to
curve the path of the rotating object,
which by Newton’s First Law will move in
a straight line unless acted upon by an
external force.
CF only appears when one considers
matters from the accelerated, non-inertial
frame of reference.
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Inertial Frame
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Accelerated Frame
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Huygens
De Vi Centrifuga, written 1859,
published after his death.
Viewed circular motion as a balance of
forces, attracting towards the center
(e.g. gravity) and “fleeing” from the
center, or centrifugal.
Arrived at correct formula for the
magnitude, Fc = mvt2/r.
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Newton
In the Principia, Newton abandoned
“balance of forces” idea in favor of the
principle of inertia (the First Law of
Motion) coupled with an inward-directed
force such as gravity to explain circular
motion.
He coined the term “centripetal force” to
drive home the distinction between his
approach and that of Huygens in dealing
with circular motion.
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Newton
Westfall points out that even Newton could not
seem to abandon CF entirely.
In illustrating his views on orbital motion,
Newton posits a small moon orbiting the earth
at the height of “the highest mountains.”
In this case, “if it were deprived of all motion
with which it proceeds in its orbit, [it would]
descend to the earth as a result of the absence
of the centrifugal force with which it had
remained in its orbit.” (Principia Book 3, Prop.
4, Scholium)
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Fictitious…Pseudo…Real
Many physics textbooks take a hard line that
CF is “fictitious” and insist that even
mentioning it in class will confuse everyone.
Others look more carefully at the reference
frame question and give CF equal treatment
with centripetal or central force.
Engineering texts present CF as real and
use it in exemplars and calculations without
apology.
A.P. French, Newtonian Mechanics, (1971)
Clear and careful discussion of
reference frames, how CF is “fictitious”
and how it can be regarded as a
legitimate “dynamical principle.”
Full section entitled “Centrifugal
Force,” another on “Centrifuges.”
Introduces in Einstein’s Principle of
Equivalence.
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Halliday and Resnik,
Fundamentals of Physics, (1970)
“Centrifugal force” is not mentioned
anywhere; there is no Index entry!
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Goldstein, Classical Mechanics, (1970)
“. . . the centrifugal force on a particle
arising from the earth’s revolution around
the sun is appreciable compared to gravity,
but it is almost exactly balanced by the
gravitational attraction to the sun. It is, of
course, just this balance between centrifugal
force and gravitational attraction that keeps
the earth . . . in orbit around the sun.”
(emphasis added)
Landau and Lifshitz, Mechanics (1976)
 Employ the Lagrangian approach to calculate the
equation of motion in a rotating frame of
reference.
 The resulting equation of motion contains two
additional terms not present in an inertial frame.
 One term represents Coriolis force, the other term
“m Ω x (r x Ω) is called the centrifugal force.”
[Ω = angular velocity vector.]
 Calculating the energy of the system, L&L arrive
at E = ½ mv2 + U + ½ m(Ω x r)2, where the last
term “is called the centrifugal potential energy.”
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Feynman, Lectures on Physics (1963)
CF is a “pseudo force” arising “due to the
fact that the observer does not have
Newton’s coordinate frame.”
“One very important feature of pseudo
forces is that they are always proportional
to the masses; the same is true of gravity.”
Pseudo forces may be connected to gravity
by general relativistic ideas.
Twigg, Science for Motor Vehicle Engineers
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British Journal Physics Education
Search on “centrifugal force” brings up
many articles and letters from 1973 to the
present.
Typical titles: “Centrifugal Force: Fact or
Fiction?” (Vol. 24, 1989),
Most recent article is by Oxford instructor
John Roche, “Introducing Motion in a
Circle,” September 2001 issue.
“There is a valid concept of centrifugal force
in physics.”
“This is the centrifugal force of physics, an
entirely fictional force.”
“I fully agree that [in teaching physics] the
fictional centrifugal force should not even be
mentioned.”
“However, if the bulk of the class intends to
go into engineering surely it would help them
to have the engineering concept of centrifugal
force explained clearly in their physics class.”
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Einstein
Theory of general relativity (GR) is based on
the Principle of Equivalence: accelerated
reference frames and gravitational fields are
physically indistinguishable.
He gives the example of an observer on a
spinning disk sensing “centrifugal force.”
Einstein asserts that this observer may
regard the disk as stationary and the force as
gravitation.
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Einstein
In Appendix III to The Meaning of
Relativity (1920 edition), Einstein
calculates the red shift of spectral lines in
a gravitational field by returning to the
spinning disk model.
He carries out the approximation using
the “potential of the centrifugal force
between the position of the [observer]
and the center of the disk.”
Einstein
The observer’s clock will then run slower
than a clock at the center by an
approximate amount:
ν = νo (1 + φ/c2)
where φ is the potential of the centrifugal
force at the observer’s location, νo is the
rate of a clock at the center of the disk,
and ν is the rate of the observer’s clock.
Because φ is negative, ν is always < νo.
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Thirring
 Hans Thirring in two seminal papers (1918 and
1921) showed that a rotating shell of matter
induces within it forces analogous to CF and
Coriolis force.
 Thirring thus suggested that CF arises because
of the rotation of the mass of fixed stars with
respect to the earth, i.e., CF is a component of
the local space-time metric.
 In his 1956 book on relativity, Pauli calls favorable
attention to the Thirring’s papers.
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Recent GR Work
Pfister and Braun, “Induction of Correct
Centrifugal Force in a Rotating Mass Shell,”
Classical and Quantum Gravitation, 2: 909918 (1985).
They carry out a new calculation of CF and
Coriolis forces inside a rotating mass shell.
Claim that their solution to Einstein’s field
equations coincides in appropriate cases to
Thirring’s result and is consistent with
“Mach’s idea concerning the relativity of
rotation.”
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Iyer and Prasanna (I&P), “Centrifugal
Force in Kerr Geometry,” Classical and
Quantum Gravitation, 10: L13-L16 (1993).
They calculate the expression for CF “at
the equatorial circumference of a rotating
body in the locally non-rotating frame of
the Kerr geometry.”
 In such a metric the sign of centrifugal
force reverses!
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Questions
Is CF “fictitious”? If the answer is yes, what
exactly is a fictitious force?
How would “force” have to be defined to
exclude CF?
Can we say that “force” can be defined only
where there is an identified physical agent
such as a magnet?
But if we so define force . . . what about the
Principle of Equivalence and Thirring?
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The next time your coffee spills onto your
slacks as your car rounds a corner too fast,
remember, the stain might be fictitious!
Classical Mechanics
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Barbour, Julian, The Discovery of Dynamics, Oxford University Press, 2001.
Cohen, I. Bernard, The Newtonian Revolution, Cambridge University Press, 1980; The
Birth of a New Physics, Norton, 1985.
Den Hartog, J.P., Mechanics, McGraw-Hill, 1948, Dover Reprint, 1961.
Dugas, René, A History of Mechanics, du Griffon 1955, Dover Reprint 1988.
Feynman, R. Leighton, R., Sands, M., Lectures on Physics, Vol. 1, Addison-Wesley, 1963.
French, A.P., Newtonian Mechanics, Norton, 1971.
Goldstein, Herbert, The Science of Mechanics, Addison-Wesley, 1950, 2nd edition 1980.
Halliday David and Resnick, Robert, Fundamentals of Physics, Wiley, 1970.
Herivel, J.W. “Newton’s Discovery of the Law of Centrifugal Force,” Isis 51: 546-553 (Dec.
1960).
Jammer, Max, Concepts of Mass, Harvard University Press, 1961, Dover Reprint 1997;
Concepts of Space, Harvard University Press, 1954, Dover Reprint 1993; Concepts of
Force, Harvard University Press, 1957, Dover Reprint 1999.
Landau, L.D. and Lifshitz, E.M., Mechanics, Pergamon, 1976.
Maxwell, J. Clerk, Matter and Motion, Thoemmes Press 1876, Dover Reprint 1991.
Meli, D., “The Relativization of Centrifugal Force,” Isis 81: 23-43 Mar. 1990).
Newton, Isaac, The Principia, translated by I. Cohen and A. Whitman, University of
California Press, 1999.
Tait, Peter and Steele, William, A Treatise on the Dynamics of Particles, MacMillan 1856,
Elibron Reprint 2003.
Westfall, Richard, Force in Newton’s Physics, Elsevier, 1971.
Teaching
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Alcaraz, A. and Ramirez, P., “Common on the ‘Role of the
Centrifugal Force in Vehicle Roll,’” American Journal of Physics
70: 5-556-557.
Roche, J., “Introducing Motion in a Circle,” Physics Education 36:
399-405.
Savage, M.D. and Williams, J.S., “Centrifugal Force: fact or Fiction?”
Physics Ed. 24: 133-140 (1989).
Stinner, A.,“Linking ‘The Book of Nature’ and ‘The Book of
Science’: Using Circular Motion as an Exemplar Beyond the
Textbook,” Science & Education 10: 323-344 (2001).
Taylor, K., “Weight and Centrifugal Force,” Physics Ed. 9: 357-360
(1974).
Twigg, Peter, Science for Motor Vehicle Engineers, ButterworthHeineman, 1996.
Ziauddin, S., “Rotational Motion and Centrifugal Force,” Physics
Education 8: 77-78 (1973).
General Relativity
Eddington, Arthur, Space, Time and Gravitation, Cambridge University
Press, 1920, reprint 1987.
Einstein, Albert, Relativity, The Special and General Theory, Barnes &
Noble, 1920, reprint 2004.
Iyer, S. and Prasanna, A.R., “Centrifugal Force in Kerr Geometry,”
Classical and Quantum Gravitation 10: L13-L16 (1993).
Mach, Ernst, The Science of Mechanics, Open Court, 1893, reprint
1989.
Mashoon, B., Hehl, F.W., Theiss, D.S., “On the Gravitational Effects of
Rotating Masses: the Thirring-Lense Papers,” General Relativity
and Gravitation 16: 711-750 (1984).
Pauli, Wolfgang, The Theory of Relativity, Pergamon, 1958, Dover
Reprint 1981.
Pfister, H and Braun, K.H., “Induction of Correct Centrifugal Force in
a Rotating Mass Shell,” Classical and Quantum Gravitation 2: 909918 (1985).
Weyl, Hermann, Space, Time, Matter , Springer, 1923, Dover Reprint,
1952.
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Sidebar: Lense-Thirring Effect
Frame Dragging and Gravity Probe B
 In 1918, Thirring joined with Josef Lense to
write another paper on general relativity, “On
the Influence of the Proper Rotation of Central
Bodies on the Motions of the Planets and
Moons According to Einstein’s Theory of
Gravitation.”
 The ideas in this paper became known as
“dragging of the inertial frames” or “framedragging.”
 Lense and Thirring conclude that this effect was
too small to be measured.
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Gravity Probe B
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Gravity Probe B uses superconducting
quantum interference devices (SQUIDs) to
measure tiny changes in the orientations of
four perfectly-spherical, quartz gyroscopes as
the experiment orbits the Earth.
The gyroscopes are housed inside a
vacuum chamber and will be maintained at
1.8 Kelvin using liquid helium.
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The probe also includes a telescope that
will be trained on a distant "guide star"
to provide a reference direction for
measurements on the gyroscopes.
General relativity predicts that the
frame-dragging effect will cause the
direction of the gyroscopes to change by
a tiny 0.041 of an arc second.
http://einstein.stanford.edu/