Centrifugal force (from Latin centrum, meaning "center", and fugere, meaning "to flee") represents the effects of inertia that arise in connection with rotation and which are experienced as an outward force away from the center of rotation. In Newtonian mechanics, the term centrifugal force is used to refer to one of two distinct concepts: an inertial force (also called a "fictitious" force) observed in a non-inertial reference frame, and a reaction force corresponding to a centripetal force.

The term is also sometimes used in Lagrangian mechanics to describe certain terms in the generalized force that depend on the choice of generalized coordinates.

The concept of centrifugal force is applied in rotating devices such as centrifuges, centrifugal pumps, centrifugal governors, centrifugal clutches, etc., as well as in centrifugal railways, planetary orbits, banked curves, etc. These devices and situations can be analyzed either in terms of the fictitious force in the rotating coordinate system of the motion relative to a center, or in terms of the centripetal and reactive centrifugal forces seen from a non-rotating frame of reference; these different forces are equal in magnitude, but centrifugal and reactive centrifugal forces are opposite in direction to the centripetal force.

History of conceptions of centrifugal and centripetal forces[link]

The conception of centrifugal force has evolved since the time of Huygens, Newton, Leibniz, and Hooke who expressed early conceptions of it. The modern conception as a fictitious force or pseudo force due to a rotating reference frame as described above evolved in the eighteenth and nineteenth centuries.

Centrifugal force has also played a role in debates in classical mechanics about detection of absolute motion. Newton suggested two arguments to answer the question of whether absolute rotation can be detected: the rotating bucket argument, and the rotating spheres argument.^[1] According to Newton, in each scenario the centrifugal force would be observed in the object's local frame (the frame where the object is stationary) only if the frame were rotating with respect to absolute space. Nearly two centuries later, Mach's principle was proposed where, instead of absolute rotation, the motion of the distant stars relative to the local inertial frame gives rise through some (hypothetical) physical law to the centrifugal force and other inertia effects. Today's view is based upon the idea of an inertial frame of reference, which privileges observers for which the laws of physics take on their simplest form, and in particular, frames that do not use centrifugal forces in their equations of motion in order to describe motions correctly.

The analogy between centrifugal force (sometimes used to create artificial gravity) and gravitational forces led to the equivalence principle of general relativity.^[2][3]

Fictitious centrifugal force[link]

Centrifugal force is often confused with centripetal force. Centrifugal force is most commonly introduced as a force associated with describing motion in a non-inertial reference frame, and referred to as a fictitious or inertial force (a description that must be understood as a technical usage of these words that means only that the force is not present in a stationary or inertial frame).^[4][5]

There are three contexts in which the concept of fictitious centrifugal force arises when describing motion using classical mechanics:^[6]

In the first context, the motion is described relative to a rotating reference frame about a fixed axis at the origin of the coordinate system. For observations made in the rotating frame, all objects appear to be under the influence of a radially outward force that is proportional to the distance from the axis of rotation and to the square of the rate of rotation (angular velocity) of the frame.

The second context is similar, and describes the motion using an accelerated local reference frame attached to a moving body, for example, the frame of passengers in a car as it rounds a corner.^[6] In this case, rotation is again involved, this time about the center of curvature of the path of the moving body. In both these contexts, the centrifugal force is zero when the rate of rotation of the reference frame is zero, independent of the motions of objects in the frame.^[7]

The third context arises in Lagrangian mechanics, and refers to a subset of generalized forces that often are not equivalent to the vector forces of Newtonian mechanics. The generalized forces are called "generalized centrifugal forces" in this context (the word generalized is sometimes forgotten). They are related to the square of the rate of change of generalized coordinates (for example, polar coordinates),^{[8][9][10][11]} used in the Lagrangian formulation of mechanics. This topic is explored in more detail below.

If objects are seen as moving from a rotating frame, this movement results in another fictitious force, the Coriolis force; and if the rate of rotation of the frame is changing, a third fictitious force, the Euler force is experienced. Together, these three fictitious forces are necessary for the formulation of correct equations of motion in a rotating reference frame.^[7]

Reactive centrifugal force[link]

A reactive centrifugal force is the reaction force to a centripetal force. A mass undergoing curved motion, such as circular motion, constantly accelerates toward the axis of rotation. This centripetal acceleration is provided by a centripetal force, which is exerted on the mass by some other object. In accordance with Newton's Third Law of Motion, the mass exerts an equal and opposite force on the object. This is the reactive centrifugal force. It is directed away from the center of rotation, and is exerted by the rotating mass on the object that originates the centripetal acceleration.^[12][13][14]

This conception of centrifugal force is very different from the fictitious force. As they both are given the same name, they may be easily conflated. Whereas the 'fictitious force' acts on the body moving in a circular path, the 'reactive force' is exerted by the body moving in a circular path onto some other object. The former is useful in analyzing the motion of the body in a rotating reference frame; the latter is useful for finding forces on other objects, in an inertial frame.

This reaction force is sometimes described as a centrifugal inertial reaction,^[15][16] that is, a force that is centrifugally directed, which is a reactive force equal and opposite to the centripetal force that is curving the path of the mass.

The concept of the reactive centrifugal force is sometimes used in mechanics and engineering. It is sometimes referred to as just centrifugal force rather than as reactive centrifugal force.^[17][18]

[edit] Fictitious vs. reactive force

The table below compares various facets of the "fictitious force" and "reactive force" concepts of centrifugal force

Example[link]

Free body diagram showing the forces on a ball and a string keeping it in circular motion. Left: inertial frame where the ball is seen to rotate. Right: co-rotating frame where the ball appears stationary. All the forces have the same magnitude, but their directions may be opposite.

The properties of the two forces in the above Table are illustrated by an example shown in the figure. The figure shows a ball in circular motion, tied to a post by a string. The post is fixed in the ground, and the string is considered too light-weight to affect the forces. The figure is an example of a free body diagram, an "exploded" engineering depiction of the different parts with the forces on each shown separately.

The forces in the inertial frame where the ball is seen to move are shown in the left column, the co-rotating frame where the ball appears not to move is shown in the right column.

The center picture of the inertial frame (left) shows the ball rotating. This circular motion departs from a straight line because the ball is subject to the centripetal radially inward force provided by the string tension. As described in the article uniform circular motion, in the case where the speed of the ball is constant, the centripetal acceleration is:

Failed to parse (Missing texvc executable; please see math/README to configure.): a = \frac{v^2}{r} \, ,

with a  the acceleration, v  the constant speed, and r  the radius of the path. The force is, of course, this acceleration multiplied by the mass of the ball.

The center picture of the co-rotating frame (right) shows the ball sitting still in a rotating frame of reference. The force on the ball due to the tension in the string is balanced by the centrifugal force introduced by the rotation of the co-rotating frame, so when the centrifugal force is included in Newton's laws of motion there is zero net force upon the ball. The appearance of a centrifugal force in this non-inertial frame is indicated in the Table, and its properties agree with those in the Table.

The lower figures show the forces upon the string, which are the same in both frames: the two ends of the string are subject to equal but oppositely directed forces. At the end of the string attached to the ball, the force is the reactive centrifugal force, the outward force exerted by the ball upon the string in reaction to the force exerted upon the ball by the tension in the string, as predicted by Newton's "action and reaction" third law of motion. As indicated in the Table, this force appears in all frames of reference, and its properties agree with those listed in the Table. This force is transmitted to the center post, where the string pulls upon the post.

At the post-end of the string, the post reacts to the pull by the string and exerts an inward directed force upon the string, labeled post reaction. The force upon the string exerted by the post balances the outward reactive centrifugal force at the other end, resulting in zero net  force upon the string. However, the two forces pulling opposite ends of the string in opposite directions place the string under tension. Detection of the non-zero tension in the string alerts the observers in the co-rotating frame that they are in fact rotating, and the ball only appears to be stationary because they are turning with it. This observation was used by Newton in his rotating spheres discussion of ways to detect absolute rotation.

Use of the term in Lagrangian mechanics[link]

Lagrangian mechanics formulates mechanics in terms of generalized coordinates {q_k}, which can be as simple as the usual polar coordinates Failed to parse (Missing texvc executable; please see math/README to configure.): (r,\ \theta)

or a much more extensive list of variables.[19][20] Within this formulation the motion is described in terms of generalized forces, using in place of Newton's laws the Euler–Lagrange equations. Among the generalized forces, those involving the square of the time derivatives {(dqk   ⁄ dt )2} are sometimes called centrifugal forces.[21][22][23][24]

The Lagrangian approach to polar coordinates that treats Failed to parse (Missing texvc executable; please see math/README to configure.): (r,\ \theta)

as generalized coordinates, Failed to parse (Missing texvc executable; please see math/README to configure.): (\dot{r},\ \dot{\theta})
as generalized velocities and Failed to parse (Missing texvc executable; please see math/README to configure.): (\ddot{r},\ \ddot{\theta})
as generalized accelerations, is outlined in another article, and found in many sources.[25][26][27] 

For the particular case of single-body motion found using the generalized coordinates Failed to parse (Missing texvc executable; please see math/README to configure.): (\dot{r},\ \dot{\theta})

in a central force, the Euler–Lagrange equations are the same equations found using Newton's second law in a co-rotating frame. For example, the radial equation is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mu\ddot{r} = \mu r\dot\theta^2 - \frac{\mathrm{d}U}{\mathrm{d}r}

where Failed to parse (Missing texvc executable; please see math/README to configure.): U(r)

is the central force potential and μ is the mass of the object. The left side is a "generalized force" and the first term on the right is the "generalized centrifugal force". However, the left side is not comparable to a Newtonian force, as it does not contain the complete acceleration, and likewise, therefore, the terms on the right-hand side are "generalized forces" and cannot be interpreted as Newtonian forces.[28]

The Lagrangian centrifugal force is derived without explicit use of a rotating frame of reference,^[29] but in the case of motion in a central potential the result is the same as the fictitious centrifugal force derived in a co-rotating frame.^[6] The Lagrangian use of "centrifugal force" in other, more general cases, however, has only a limited connection to the Newtonian definition.

References[link]

Julius Sumner Miller (May 17, 1909 – April 14, 1987) was an American physicist and television personality. He is best known for his work on children's television programs in North America and Australia.

Off-screen[link]

Julius Sumner Miller was born in Billerica, Massachusetts the youngest of nine children. His father was Latvian, his Lithuanian mother spoke 12 languages.^[1][2]

Miller graduated with a Philosophy degree and a Master's in Physics from Boston University in 1933 but due to the Depression worked as a butler for a wealthy Boston doctor for the next two years during which time he married the doctor's maid, Alice Brown; they were to have no children, but he was to go on to reach millions of children through his popular science programs.^[2]

After making over 700 applications for employment, in 1937 he was offered a place in the Physics Department of Dillard University, a private, black liberal arts college in New Orleans. During World War II he worked as a civilian physicist for the U.S. Army Signal Corps while holding fellowships in physics at the universities of Idaho and Oklahoma and was a Ford Foundation Fellow at UCLA. In 1950 he enrolled in the Institute for Advanced Studies in Princeton, N.J. where he was a student of Albert Einstein. He became a lifelong friend of Einstein and went on to amass a collection of Einstein memorabilia that included Einstein's birth certificate. Miller also taught at Princeton but disliked large institutions, leaving in 1952 to join the Physics Department at the then small El Camino College in Torrance, California (1952–1974), to maximum student enrollments due to his great popularity^[2] and where he was instantly recognizable by his casual hair and horn-rimmed spectacles.

Miller was intolerant of misspelled words and misplaced punctuation and often angered his colleagues because he charged that the students of most faculty were not learning enough. During an interview in the 1940s, he stated that intellectual life in America was in trouble, a belief he held for the rest of his life.

"We are approaching a darkness in the land. Boys and girls are emerging from every level of school with certificates and degrees, but they can't read, write or calculate. We don't have academic honesty or intellectual rigor. Schools have abandoned integrity and rigor."^[2]

From 1963 to 1986 he was the visiting lecturer for the Physics Department of the University of Sydney ^[3] and from 1965 to 1985, the US Air Force Academy.

Television[link]

Miller began hosting his educational program, "Why Is It So?" in 1959 on KNXT Channel 2 in Los Angeles. From 1962 to 1964, Miller was Disney's "Professor Wonderful" on new introductions, filmed at Disneyland, to the syndicated reruns of The Mickey Mouse Club. During the same period, he appeared on a semi-regular basis, performing physics experiments, on Steve Allen's late night TV show in Hollywood, syndicated by Group W.

Miller's first television appearance in Australia was on Bob Sanders People in 1963. In an improvised physics demonstration he attempted to drive a straw through a raw potato. A paper straw does not have the strength but if you pinch the end, the trapped air acts as a piston, easily piercing the potato. For the first time in his career he could not get it to work and loudly exclaimed "Australian straws ain't worth a damn". The next morning, Miller arrived at his Sydney University laboratory to find one million drinking straws on the floor with a telegram reading "You might find one of these fitting your requirements". He later stated "I sat amongst the straws with straws stuck in my hair and ears. But clearly I had made a mistake. I should have said: "Australian potatoes ain’t worth a damn," and I’d have cornered the potato market!"

Shortly after, he was offered a job presenting science for ABC Television. When asked how much money he wanted he replied he never asked, he listened to an offer then "multiplied it by a factor between two and ten". Due to budget constraints the offer was withdrawn but an agreement was reached for Miller to host his own science based TV series which was filmed at the University of Sydney where he taught. Why Is It So? (his eventual stock phrase), was broadcast from 1963 to 1986 and became an instant hit known for its "cool experiments, interesting science, and fantastic hair." The 1960s program became Demonstrations in Physics (also called Science Demonstrations when it was aired on American Public television). He introduced each episode with the line: "How do you do, ladies and gentlemen, and boys and girls [sometimes adding "and teachers"]. I am Julius Sumner Miller, and physics is my business."

"My first TV series on demonstrations in physics - titled Why Is It So were now seen and heard over the land. The mail was massive. The academics were a special triumph for me. They charged me with being superficial and trivial. If I had done what they wanted my programs would be as dull as their classes! I knew my purpose well and clear: to show how Nature behaves without cluttering its beauty with abtruse mathematics. Why cloud the charm of a Chladni plate with a Bessel function?"^[4]

Miller's on-air popularity was due to an enthusiasm not normally associated with serious science. Shows would be liberally sprinkled with phrases such as "He who is not stirred by the beauty of it is already dead!" and he also liked to trick the audience. A common ploy would be to hold up an empty glass and ask guests to confirm it was empty....then chide them for not noticing it was full of air. Before each demonstration he would usually ask for a show of hands to indicate which of several results they expected. Often he would then add "hands up those who don’t care".

In 1964 Miller suffered a near fatal heart attack. Scheduled to give a lecture in Australia, he sent Sydney University a telegram saying; "I've dropped dead here." He would suffer a second heart attack in 1986.^[2]

In 1966 questions from his show with an answer to the previous day’s question were published as "Millergrams" for The Australian newspaper. 112 of these questions were published as a book "Millergrams; Some Enchanting Questions for Enquiring Minds". Further books were published in 1967: "The Second Book of Millergrams: Some More Enchanting Questions for Enquiring Minds" and 1988: "Why is it So?: The Very Best Millergrams of Professor Julius Sumner Miller".

Sample Millergram:
Q32: A juggler comes to a foot-bridge of rather flimsy design. He has in hand four balls. The maximum load is no more than the juggler himself and one ball. Can he get across the bridge by juggling the balls, always having at most one ball in the hand (and three in the air)?
A: No. A falling ball exerts a force on the hand greater than its own weight.
Rather, a 'thrown' ball exerts greater force than a 'held' one. That is, the additional force equal and opposite to that imparted to a flung ball, in addition to the juggler's mass, would exceed the bridge's tolerance (the bridge can tolerate a juggler and held ball, but not the additional downward force associated with forcing a ball 'up').

Miller appeared as "The Professor" in the Canadian series The Hilarious House of Frightenstein (1971), in a 2- to 3-minute segment each episode where he demonstrated physics experiments and explained the principles involved.

Miller was also an occasional guest in the 1970s on The Tonight Show Starring Johnny Carson in the United States.

Commercials[link]

During the 1980s Miller appeared in a famous series of Australian television commercials for Cadbury chocolate, using his stock phrase "Why is it so?", demonstrating a simple scientific principle, and describing how each block of chocolate "embraces substantial nourishment and enjoyment," and contained "a glass and a half of full-cream dairy milk."

While in Australia, Miller also appeared in ads for non-stick saucepans and Ampol Petroleum which included demonstrations of real principles of physics, albeit briefly. The ads were sufficiently popular to be played for some years after his death.

Foundations[link]

In February 1987 Miller became ill while visiting Australia and returned to the United States where he was diagnosed with leukemia. Miller died six weeks later on April 14, 1987, in San Jose, California. Miller willed his body to the University of Southern California's School of Dentistry, no services were held at his request.^[2] Professor Miller's wife, Alice Brown Miller, wanted to perpetuate the memory and achievements of her husband, and so conceived the idea of the Julius Sumner Miller Foundation, which was established in 1998.^[1]

Through an offer by Cadbury-Schweppes Pty Ltd, the Cadbury-Julius Sumner Miller Scholarship for Academic Excellence was set up to provide undergraduate scholarships in the School of Physics at the University of Sydney.

In 1993 the Australian Science Foundation for Physics established the Julius Sumner Miller Fellowship in his memory. The fellowship is currently held by Karl Kruszelnicki, best known as "Doctor Karl" for his appearances on Australian radio and television as a science commentator and author.

Popular culture[link]

In 1966 Miller devised a word game "Milleranagrams" which was published in Australia by John Sands Limited. The game, whose only materials were 200 Scrabble-like letter tiles, required players in their turn to draw an unseen tile from the "pool" then either make a word from their stock of tiles or add a tile to a word already on the table (rearrangement of the letters being permitted, hence the name) to form another word.

Miller remains popular in Australia, where he is still quoted. As Professor Julius Sumner Miller often asked, "Why is it so?" and its variations are still often used in newspaper articles that pose questions unrelated to science.^[5]

Bibliography[link]

• Millergrams; Some Enchanting Questions for Enquiring Minds, Ure Smith, 1966
• The Second Book of Millergrams: Some More Enchanting Questions for Enquiring Minds, Ure Smith, 1967
• Quiz Questions in Physics, Horwitz-Martin, Australia 1967
• Why It Is So, ABC books, 1971 ISBN 0-642-97296-6
• The Kitchen Professor, ABC books, 1972 ISBN 0-642-97352-0
• Why It Is So: Heat and Temperature, ABC books, 1973 ISBN 0-642-97496-9
• Why It Is So: Sound and Electricity & Magnetism, ABC books, 1973 ISBN 0-642-97584-1
• Why It Is So: Mechanics, Heat & Temperature, Sound and Electricity, ABC books, 1978 ISBN 0-642-97523-X
• Enchanting Questions for Enquiring Minds, Currey/O'Neil, 1982 ISBN 0-85902-280-3
• Why is it so?: the very best Millergrams of Professor Julius Sumner Miller, Australian Red Cross Society, Ringwood, Vic; Penguin Books, 1988
• The Days of My Life: an autobiography, Macmillan, 1989. ISBN 0-333-50337-6

References[link]

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External links[link]

Wikiquote has a collection of quotations related to: Julius Sumner Miller

• Julius Sumner Miller Foundation
• Australian Julius Sumner Miller tribute page
• Julius Sumner Miller page at a Hilarious House of Frightenstein tribute site
• Julius Sumner Miller at TV.com
• Julius Sumner Miller at the Internet Movie Database
• Why is it so? site at Australia's ABC.net.au featuring original episodes
• http://www.physics.usyd.edu.au/foundation/Outreach/DRK/jsm.pdf Julius Sumner Miller 1909 - 1987, Physics Department, University of Sydney.