# User:FyzixFighter/Centrifugal force/draft

Centrifugal force (from Latin centrum "center" and fugere "to flee") can generally be any force directed outward relative to some origin. More particularly, in classical mechanics, centrifugal force is an outward force which arises when describing the motion of objects in a rotating reference frame. Because a rotating frame is an example of a non-inertial reference frame, Newton's laws of motion cannot be used to describe the dynamics within the frame. However, a rotating frame can be treated as if it were an inertial frame where Newton's laws are valid if so-called fictitious forces (also known as inertial or pseudo- forces) are included in the sum of external forces on an object. The centrifugal force is what is usually thought of as the cause for apparent outward movement like that of passengers in a vehicle turning a corner, of the weights in a centrifugal governor, and of particles in a centrifuge. From the standpoint of an observer in an inertial frame, the effects can be explained as results of inertia without invoking the centrifugal force. Centrifugal force should not be confused with centripetal force or the reactive centrifugal force, both of which are real forces independent of the frame of the observer.

Analysis of motion within rotating frames can be greatly simplified by the use of the fictitious forces. By starting with an inertial frame, where Newton's laws of motion hold, and seeing how the time derivatives of a position vector change when transforming to a rotating reference frame, the various fictitious forces and their forms can be identified. Rotating frames and fictitious forces can often reduce the description of motion in two dimensions to a simpler description in one dimensions (corresponding to a co-rotating frame). In this approach, circular motion in an inertial frame, which only requires the presence of a centripetal force, becomes the balance between the real centripetal force and the frame-determined centrifugal force in a rotating frame where the object appears stationary. Also in this approach, if a rotating frame is chosen so that just the angular position of an object is held fixed, more complicated radial motion, like that of elliptical and open orbits, appears when the centripetal and centrifugal forces do not balance. The general approach however is not limited to these co-rotating frames, but can be equally applied to objects at motion in any rotating frame. All objects in any rotating frame will appear to experience the outward centrifugal force.

## History

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.

• needs better summary of history article - more than two sentences

## In classical Newtonian mechanics

Although Newton's laws of motion hold exclusively in inertial frames, often times is far more convenient and more advantageous to describe the motion of objects within a rotating reference frame.[1][2] Sometimes the calculations are simpler (an example is inertial circles), and sometimes the intuitive picture coincides more closely with the rotational frame (an example is sedimentation in a centrifuge). By treating the extra acceleration terms due to the rotation of the frame as if they were forces, subtracting them from the physical forces, it's possible to treat the second time derivative of position (relative to the rotating frame) as absolute acceleration. Thus the analysis using Newton's laws of motion can proceed as if the reference frame was inertial, provided the fictitious force terms are included in the sum of external forces.[3] For example, centrifugal force is used in the FAA pilot's manual in describing turns.[4] Other examples are such systems as planets, centrifuges, carousels, turning cars, spinning buckets, and rotating space stations.[5][6][7]

A disadvantage of a rotating reference frame is that it can be more difficult to apply special relativity (for example, from the perspective of the Earth the stars seem to traverse many light-years each day). It is possible to do so if a metric tensor is introduced, but the speed of light may not be constant and clocks within the frame are not synchronized.[8][9]

There are three general scenarios in which this concept of a fictitious centrifugal force arises when describing motion:[10]

1. When 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.
2. When the motion is described 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.[10] In this case, rotation is again involved, this time about the center of curvature of the path of the moving body. The first context can be seen as a special scenario within this second context in which the origin of the coordinate system and the axis of rotation are always coincident. In both, 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.[11]
3. The third context is the most general, and subsumes the first two, as well as stationary curved coordinates (e.g., polar coordinates).[12][13][14][15] The centrifugal force appears when the terms for the radial component of the equation of motion are rearranged to resemble Newton's second law for one-dimensional motion. Therefore, the centrifugal force is simply the sign-reversal of the centripetal acceleration for motion along curves where the radial distance is fixed and is related to the Christoffel symbol term related to that curvature. While no rotation is necessary in this derivation, reapplying Newtonian definitions of force and acceleration to the rearranged equation necessarily implies observing the motion from a co-rotating frame of reference.[16][17]

In each of these scenarios, the centrifugal force is an inertial force used for convenience and implied by a specific, non-inertial reference frame.

If objects are seen as moving within 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.[18] Together, these three fictitious forces allow for the creation of correct equations of motion in a rotating reference frame.[11]

### Derivation

In a rotating frame of reference the time derivatives of any position vector r, such as the velocity and acceleration vectors, will differ from the time derivatives in an inertial frame according to the frame's rotation. The first time derivative [dr/dt] evaluated from a reference frame with a coincident origin at ${\displaystyle r=0}$ but rotating with the absolute angular velocity Ω is:[19]

${\displaystyle {\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}=\left[{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\right]+{\boldsymbol {\Omega }}\times {\boldsymbol {r}}\ ,}$

where ${\displaystyle \times }$ denotes the vector cross product and square brackets […] denote evaluation in the rotating frame of reference. In other words, the apparent velocity in the rotating frame is altered by the amount of the apparent rotation ${\displaystyle {\boldsymbol {\Omega }}\times {\boldsymbol {r}}}$ at each point, which is perpendicular to both the vector from the origin r and the axis of rotation Ω and directly proportional in magnitude to each of them. The vector Ω has magnitude Ω equal to the rate of rotation and is directed along the axis of rotation according to the right-hand rule.

Newton's law of motion for a particle of mass m written in vector form is

${\displaystyle {\boldsymbol {F}}=m{\boldsymbol {a}}\ ,}$

where F is the vector sum of the physical forces applied to the particle and a is the absolute acceleration[20] of the particle, given by:

${\displaystyle {\boldsymbol {a}}={\frac {\operatorname {d} ^{2}{\boldsymbol {r}}}{\operatorname {d} t^{2}}}\ ,}$

where r is the position vector of the particle.

By twice applying the transformation above from the inertial to the rotating frame, the absolute acceleration of the particle can be written as:

{\displaystyle {\begin{aligned}{\boldsymbol {a}}&={\frac {\operatorname {d} ^{2}{\boldsymbol {r}}}{\operatorname {d} t^{2}}}={\frac {\operatorname {d} }{\operatorname {d} t}}{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}={\frac {\operatorname {d} }{\operatorname {d} t}}\left(\left[{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\right]+{\boldsymbol {\Omega }}\times {\boldsymbol {r}}\ \right)\\&=\left[{\frac {\operatorname {d} ^{2}{\boldsymbol {r}}}{\operatorname {d} t^{2}}}\right]+{\frac {\operatorname {d} {\boldsymbol {\Omega }}}{\operatorname {d} t}}\times {\boldsymbol {r}}+2{\boldsymbol {\Omega }}\times \left[{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\right]+{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times {\boldsymbol {r}})\ .\end{aligned}}}

The apparent acceleration in the rotating frame is [d2r/dt2]. An observer unaware of the rotation would expect this to be zero in the absence of outside forces. However Newton's laws of motion apply only in the inertial frame and describe motion in terms of the absolute acceleration d2r/dt2. Therefore the observer perceives the extra terms as contributions due to fictitious forces. These terms in the apparent acceleration are independent of mass; so it appears that each of these fictitious forces, like gravity, pulls on an object in proportion to its mass. When these forces are added, the equation of motion has the form:[21][22][23]

${\displaystyle {\boldsymbol {F}}-m{\frac {\operatorname {d} {\boldsymbol {\Omega }}}{\operatorname {d} t}}\times {\boldsymbol {r}}-2m{\boldsymbol {\Omega }}\times \left[{\frac {\operatorname {d} \mathbf {r} }{\operatorname {d} t}}\right]-m{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times {\boldsymbol {r}})}$${\displaystyle =m\left[{\frac {\operatorname {d} ^{2}{\boldsymbol {r}}}{\operatorname {d} t^{2}}}\right]\ .}$

From the viewpoint of the rotating frame, the additional force terms are experienced just like the real external forces and contribute to the apparent acceleration.[24][25] The additional terms on the force side of the equation can be recognized as, reading from left to right, the Euler force, the Coriolis force, and the centrifugal force, respectively. Unlike the other two fictitious forces, the centrifugal force always points directly away from the axis of rotation of the rotating reference frame, with magnitude mΩ2r, and unlike the Coriolis force in particular, the centrifugal force is independent of the motion of the particle in the rotating frame. As expected, for a non-rotating inertial frame of reference ${\displaystyle ({\boldsymbol {\Omega }}=0)}$ the centrifugal force and all other fictitious forces disappear.[26]

### Stationary objects

A simple example of body observed to be moving in a rotating frame is that of a body that is stationary relative to a non-rotating inertial. Viewed from the rotating frame, the body follows a circular path and therefore, by application of Newton's laws to what looks like circular motion in the rotating frame at a radius r, requires an inward force of −m Ω2 r, where Ω is angular rate of rotation of the frame. This centripetal force in the rotating frame is provided as a net fictitious force that is the sum of the radially outward centrifugal force m Ω2 r and the Coriolis force −2m Ω × vrot.[27] The magnitude of the Coriolis force depends on the velocity as seen in the rotating frame, vrot, which is given by −Ω × r.[28] This results in a Coriolis force with a value −2m Ω2 r, the negative sign indicating that it is radially inward. The combination of the centrifugal and Coriolis force is then m Ω2 r − 2m Ω2 r = −m Ω2 r, exactly the centripetal force required by Newton's laws for circular motion.[29]

## In Lagrangian mechanics

Lagrangian mechanics formulates mechanics in terms of generalized coordinates ${\displaystyle \{q_{k}\}}$, which can be as simple as the usual polar coordinates ${\displaystyle (r,\ \theta )}$ or a much more extensive list of variables. 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 ${\displaystyle \{(\mathrm {d} q_{k}/\mathrm {d} t)^{2}\}}$ are sometimes called centrifugal forces.[30][31][32]

For the particular case of single-body motion found using the polar coordinates ${\displaystyle ({\dot {r}},\ {\dot {\theta }})}$ as the generalized coordinates 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:

${\displaystyle \mu {\ddot {r}}=\mu r{\dot {\theta }}^{2}-{\frac {\mathrm {d} U}{\mathrm {d} r}}}$

where ${\displaystyle U(r)}$ 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.[33] In the special case of motion in a central potential described with polar coordinates, the Lagrangian centrifugal force is the same as the fictitious centrifugal force derived in a co-rotating frame. However, the Lagrangian use of "centrifugal force" in other, more general cases has only a limited connection to the Newtonian definition.

## Absolute rotation

Three scenarios were suggested by Newton to answer the question of whether the absolute rotation of a local frame can be detected; that is, if an observer can decide whether an observed object is rotating or if the observer is rotating.[34]

• The shape of the surface of water rotating in a bucket. Centrifugal force will force the surface of the water to be concave.
• The tension in a string joining two spheres rotating about their center of mass. The tension in the string will be proportional to the centrifugal force on each sphere as it rotates around the common center of mass.
• The oblateness of a sphere of freely flowing material. The oblate spheroid shape reflects, following Clairaut's theorem, the balance between containment by gravitational attraction and dispersal by centrifugal force.

In these scenarios, the effects attributed to centrifugal force are only observed in the local frame, that is the frame in which the objects are stationary, if the objects are undergoing absolute rotation relative to an inertial frame. By contrast, in an inertial frame, the observed effects arise as a consequence of the inertia of the objects without the need to introduce a centrifugal force.

## Applications

The operations of numerous common rotating mechanical systems are most easily conceptualized in terms of centrifugal force. For example:

• A centrifugal governor regulates the speed of an engine by using spinning masses that move radially, adjusting the throttle, as the engine changes speed. In the reference frame of the spinning masses, centrifugal force causes the radial movement.
• A centrifugal clutch is used in small engine-powered devices such as chain saws, go-karts and model helicopters. It allows the engine to start and idle without driving the device but automatically and smoothly engages the drive as the engine speed rises. Inertial drum brake ascenders used in rock climbing and the inertia reels used in many automobile seat belts operate on the same principle.
• Centrifugal forces can be used to generate artificial gravity, as in proposed designs for rotating space stations. The Mars Gravity Biosatellite will study the effects of Mars-level gravity on mice with gravity simulated in this way.
• Spin casting and centrifugal casting are production methods that uses centrifugal force to disperse liquid metal or plastic throughout the negative space of a mold.
• Centrifuges are used in science and industry to separate substances. In the reference frame spinning with the centrifuge, the centrifugal force induces a hydrostatic pressure gradient in fluid-filled tubes oriented perpendicular to the axis of rotation, giving rise to large buoyant forces which push low-density particles inward. Elements or particles denser than the fluid move outward under the influence of the centrifugal force. This is effectively Archimedes' principle as generated by centrifugal force as opposed to being generated by gravity.
• Some amusement park rides make use of centrifugal forces. For instance, a Gravitron’s spin forces riders against a wall and allows riders to be elevated above the machine’s floor in defiance of Earth’s gravity.

Nevertheless, all of these systems can also be described without requiring the concept of centrifugal force, in terms of motions and forces in an inertial frame, at the cost of taking somewhat more care in the consideration of forces and motions within the system.

## Notes and references

1. ^ Stephen T. Thornton & Jerry B. Marion (2004). Classical Dynamics of Particles and Systems (5th ed.). Belmont CA: Brook/Cole. Chapter 10. ISBN 0534408966. line feed character in |author= at position 22 (help)
2. ^ John Robert Taylor (2004). Classical Mechanics. Sausalito CA: University Science Books. Chapter 9, pp. 327 ff. ISBN 189138922X.
3. ^ Robert Resnick & David Halliday (1966). Physics. Wiley. p. 121. ISBN 0471345245.
4. ^ Federal Aviation Administration (2007). Pilot's Encyclopedia of Aeronautical Knowledge. Oklahoma City OK: Skyhorse Publishing Inc. Figure 3–21. ISBN 1602390347.
5. ^ Richard Hubbard (2000). Boater's Bowditch: The Small Craft American Practical Navigator. NY: McGraw-Hill Professional. p. 54. ISBN 0071361367.
6. ^ Lawrence K. Wang, Norman C. Pereira (1979). Handbook of Environmental Engineering: Air and Noise Pollution Control. Humana Press. p. 63. ISBN 0896030016.
7. ^ Lee M. Grenci, Jon M. Nese (2001). A World of Weather: Fundamentals of Meteorology. Kendall Hunt. p. 272. ISBN 0787277169.
8. ^ Philip Gibbs. "Physics FAQ". University of California Riverside.
9. ^ T.A.Weber. "Measurements on a rotating frame in relativity, and the Wilson and Wilson experiment" (PDF). Am J Phys. pp. 946–953.
10. ^ a b See p. 5 in Donato Bini, Paolo Carini, Robert T Jantzen (1997). "The intrinsic derivative and centrifugal forces in general relativity: I. Theoretical foundations" (PDF). International Journal of Modern Physics D. 6 (1).. The companion paper is Donato Bini, Paolo Carini, Robert T Jantzen (1997). "The intrinsic derivative and centrifugal forces in general relativity: II. Applications to circular orbits in some stationary axisymmetric spacetimes" (PDF). International Journal of Modern Physics D. 6 (1).
11. ^ a b Fetter, Alexander L. and John Dirk Walecka (2003). Theoretical Mechanics of Particles and Continua (Reprint of McGraw-Hill 1980 ed.). Courier Dover Publications. ISBN 0486432610, pp. 38–39.
12. ^ "Classical Dynamics of Particles and Systems", Marion and Thornton, Brooks Cole; 5 edition 2003.
13. ^ "Methods of Applied Mathematics" By Francis B. Hildebrand, 1992, Dover, p 156.
14. ^ "Statistical Mechanics" By Donald Allan McQuarrie, 2000, University Science Books.
15. ^ "Essential Mathematical Methods for Physicists" By Hans-Jurgen Weber, George Brown Arfken, Academic Press, 2004, p 843.
16. ^ Atam Parkash Arya (1990). Introduction to Classical Mechanics. Allyn and Bacon. p. 231. ISBN 0205120288.
17. ^ Jeremy B. Tatum "Celestial Mechanics" Chapter 16
18. ^ Jerrold E. Marsden, Tudor S. Ratiu (1999). Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems. Springer. p. 251. ISBN 038798643X.
19. ^ John L. Synge (2007). Principles of Mechanics (Reprint of Second Edition of 1942 ed.). Read Books. p. 347. ISBN 1406746703.
20. ^ The term "absolute" meaning as seen in any inertial frame of reference; for example "absolute acceleration" or "absolute derivative".
21. ^ John R Taylor (2005). Classical Mechanics. University Science Books. p. 342. ISBN 1-891389-22-X.
22. ^ LD Landau and LM Lifshitz (1976). Mechanics (Third ed.). Oxford: Butterworth-Heinemann. p. 128. ISBN 978-0-7506-2896-9.
23. ^ Louis N. Hand, Janet D. Finch (1998). Analytical Mechanics. Cambridge University Press. p. 267. ISBN 0521575729.
24. ^ Mark P Silverman (2002). A universe of atoms, an atom in the universe (2 ed.). Springer. p. 249. ISBN 0387954376.
25. ^ John Robert Taylor (2005). op. cit.. Sausalito, Calif.: Univ. Science Books. p. 329. ISBN 189138922X.
26. ^ Morton Tavel (2002). Contemporary Physics and the Limits of Knowledge. Rutgers University Press. p. 93. ISBN 0813530776. Noninertial forces, like centrifugal and Coriolis forces, can be eliminated by jumping into a reference frame that moves with constant velocity, the frame that Newton called inertial.
27. ^ Georg Joos & Ira M. Freeman (1986). Theoretical Physics. New York: Courier Dover Publications. p. 233. ISBN 0486652270.
28. ^ The vector cross product of the two orthogonal vectors Ω and r is a vector of magnitude equal to the product of their magnitudes, namely Ω r = vrot, and with direction given by the right-hand rule, in this case found by aligning the thumb with Ω, the index finger with r, and the middle finger normal to these two fingers points in the direction of −vrot.
29. ^ Burgel, B. (1967). "Centrifugal Force". American Journal of Physics. 35: 649. doi:10.1119/1.1974204.
30. ^ Shuzhi S. Ge, Tong Heng Lee, Christopher John Harris (1998). Adaptive Neural Network Control of Robotic Manipulators. World Scientific. pp. 47–48. ISBN 981023452X. In the above Euler–Lagrange equations, there are three types of terms. The first involves the second derivative of the generalized co-ordinates. The second is quadratic in ${\displaystyle {\boldsymbol {\dot {q}}}}$ where the coefficients may depend on ${\displaystyle {\boldsymbol {q}}}$. These are further classified into two types. Terms involving a product of the type ${\displaystyle {{\dot {q}}_{i}}^{2}}$ are called centrifugal forces while those involving a product of the type ${\displaystyle {\dot {q}}_{i}{\dot {q}}_{j}}$ for i ≠ j are called Coriolis forces. The third type is functions of ${\displaystyle {\boldsymbol {q}}}$ only and are called gravitational forces.
31. ^ R. K. Mittal, I. J. Nagrath (2003). Robotics and Control. Tata McGraw-Hill. p. 202. ISBN 0070482934.
32. ^ T Yanao & K Takatsuka (2005). "Effects of an intrinsic metric of molecular internal space". In Mikito Toda, Tamiki Komatsuzaki, Stuart A. Rice, Tetsuro Konishi, R. Stephen Berry. Geometrical Structures Of Phase Space In Multi-dimensional Chaos: Applications to chemical reaction dynamics in complex systems. Wiley. p. 98. ISBN 0471711578. As is evident from the first terms …, which are proportional to the square of ${\displaystyle {\dot {\phi }}}$, a kind of "centrifugal force" arises … We call this force "democratic centrifugal force". Of course, DCF is different from the ordinary centrifugal force, and it arises even in a system of zero angular momentum.
33. ^ Henry M. Stommel and Dennis W. Moore (1989). An Introduction to the Coriolis Force. Columbia University Press. pp. 36–38.
34. ^ Louis N. Hand, Janet D. Finch (1998). Analytical Mechanics. Cambridge University Press. p. 324. ISBN 0521575729. and I. Bernard Cohen, George Edwin Smith (2002). The Cambridge companion to Newton. Cambridge University Press. p. 43. ISBN 0521656966.

## Draft notes to self

• Consolidate multiple uses of the same reference
1. Taylor, "Classical Mechanics"
2. Hand and Finch, "Analytical Mechanics"
3. Thornton and Marion, "Classical Dynamics of Particles and Systems"
• Reduce strings of 3+ references for simple statements
• really don't like the long list at the end of the lede, most statements in lede don't need to be sourced since the lede acts as a summary of the articles - reduce to a few or move them to the body?
• Which examples to keep?
• Intro needs to be rewritten
• better summary of article and better layman's explanation