Reactive centrifugal force

A reactive centrifugal force is the reaction force to a centripetal force.

A mass undergoing circular motion constantly accelerates toward the axis of rotation. This centripetal acceleration is caused by a centripetal force, which is exerted on the mass by some other object. In accordance with , the mass exerts an equal and opposite force on the object. This is the "real" or "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.cite book |title=Inside relativity |author=Delo E. Mook & Thomas Vargish |page=p. 47 |url=http://books.google.com/books?id=QnJqIyk_dzIC&pg=PA47&dq=%22reactive+centrifugal+force%22&lr=&as_brr=0&sig=EDmHHDZRZB4AC37tklWe03SD_tY
isbn=0691025207|publisher=Princeton University Press|location=Princeton NJ |year=1987] [http://physnet.org/modules/pdf_modules/m17.pdf "Acceleration and force in circular motion" by Peter Signell] ]

Although this force was simply called "centrifugal force" by Isaac Newton, [Isaac Newton, [http://members.tripod.com/~gravitee/booki2.htm "Principia" (1687)] , Book I, Section II, Scholium following Proposition IV] , in modern language that phrase most often refers to the "fictitious" centrifugal force, which is different.

Reactive centrifugal force

Figure 1 (upper left) shows a ball in uniform circular motion held to its path by a massless string tied to a post stuck in the ground. Because the string is assumed to be massless, no centripetal force is needed to maintain the circular motion of the string itself. Newton's second law requires that any body not moving in a straight line is subject to a force, and indeed, panel 2 shows the ball is subjected to a centripetal force by the string. Newton's third law states that if the string exerts a centripetal force on the ball, the ball will exert an equal reaction upon the string and post, the reactive centrifugal force shown in panel 3. Panel 4 shows the string and ball combined, both in uniform circular motion. The centripetal force of panel 2 is applied by the post to the end of the string. The string is looked at separately in panel 5, and is subject to a reactive centrifugal force at one end balanced by a centripetal force at the other end, placing the string under "tension". Panel 6 shows the post separately. Due to Newton's third law, the post is subject to the reactive centrifugal force, a reaction to the centripetal force of panel 4, and also is subject to the centripetal force exerted by the post-hole on the post. The net force on the post is zero, so it does not move.

Example: The turning car

Here is a more picturesque example: A car with a passenger inside driving around a curve.cite book |title=Newnes Mechanical Engineer's Pocket Book|page=p. 111|url=http://books.google.com/books?id=IZdDk1cMlLcC&pg=PA111&dq=reactive+%22centrifugal+force%22&lr=&as_brr=0&sig=fg72IYE6vZxgRDWEdTikY5780LE
author=Roger Leslie Timings|isbn=0750665084|year=2005|publisher=Elsevier/Newnes|location=Oxford] See Figure 2. The road exerts a centripetal force upon the car, forcing the car to make the turn. This force is called a centri"petal" ("center seeking") force because its vector changes direction to continue to point toward the center (precisely, the center of curvature) of the car's arc as the car traverses it.

Viewed from an inertial frame of reference, the passenger's inertia keeps the passenger moving with constant speed and direction as the car begins to turn; that is, failing interference by the car, the passenger is going to travel a straight line, not go around the turn. From this point of view, the passenger is not pushed toward the outside of the path the car follows; instead, the passenger proceeds on a straight path and the car's path curves to meet the passenger. If the seat is slippery, the passenger slides along a straight path across the car seat until meeting the car door, whereupon the door pushes the passenger to follow 'round the curve.

If the car seat is not slippery, it applies a sideways force on the passenger to accelerate around the turn with the car.

If the car is acting upon the passenger, then the passenger must be acting upon the car with an equal and opposite force (). See Figure 3. Being opposite, this reaction force is directed "away" from the center, therefore centri"fugal". It is critical to realize that this centrifugal force acts upon the car seat, not upon the passenger. [http://www.infoplease.com/ce6/sci/A0811114.html The Columbia Electronic Encyclopedia: "Centripetal force and centrifugal force"] ]

Figure 3 also shows that the car seat is subjected to a centripetal force applied by the car itself. This centripetal force exceeds the reactive centrifugal force exerted on the seat by the passenger, causing the seat to move around the turn under a net inward force.

Figure 3 shows the car body is subject to the centripetal force from the road of Figure 1, and also is subject to the reactive centrifugal force exerted by the car seat. The inward force on the car body exceeds the outward centrifugal force due to the cart seat by enough to make the car body follow the turn.

The sketch in Figure 3 shows the forces are not colinear, so they exert a torque on the various parts. Although torque is a likely real-world consideration (the car tends to tip over as it rounds the turn), it simply is ignored here to focus upon the reactive centrifugal force.

The "centrifugal reaction force" with which the passenger pushes back against the car seat is given by:

:$mathbf\{F\}\_mathrm\{centrifugal\}$ $=\; -\; m\; mathbf\{a\}\_\{mathrm\{centripetal$::::$=\; m\; omega^2\; mathbf\{r\}\_perp\; ,$

where $m$ is the mass of the passenger, $omega$ the rotational speed (in radians per unit time), and r_$perp$ the radius vector outward from the axis of rotation to the passenger.

The reactive centrifugal force is a real force, but the term is rarely used in modern discussions.

Confusion and misconceptions

Centrifugal force can be a confusing term because it is used (or misused) in more than one instance, and because sloppy labeling can obscure which forces are acting upon which objects in a system. When diagramming forces in a system, one must describe each object separately, attaching only those forces acting "upon" it (not forces that it "exerts" upon other objects). See Figures 1, 3 and 4 for examples.

Applications

Reactive centrifugal force is a real force that can be used to do physical work. A few examples:
* A centrifugal governor regulates the speed of an engine by using spinning masses set in circular motion by the engine. Should the engine speed increase, the masses swing out to a larger radius and trigger a cut in the throttle. The net force on each spinning mass is the resultant of the downward force of gravity and the tension from the rod or cable supporting the mass. See Figure 4. This net force provides the centripetal force necessary to maintain the circular orbit of the mass. The spinning mass exerts a reaction force on the support rod. This reaction force (directed along the rod) has a horizontal component that is the reactive centrifugal force exerted by the circling mass, and a vertical component. See Figure 1 for a top-down view. If a change in speed causes the spinning mass to become more elevated, the work done is stored as potential energy in the gravitational field. However, none of this work due to this vertical displacement is done by the reactive centrifugal force because it is horizontal.
* 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. The idea is similar to the centrifugal governor discussed above, but in place of gravity a spring is used to support the spinning clutch shoe. The spring provides the centripetal force to the shoe, which moves to larger radius as the speed increases, ultimately engaging the driveshaft . The force of tension applied to the spring by the spinning shoe is the reactive centrifugal force. This tension extends the spring, so the reactive centrifugal force does work stored as energy in the extended spring.

Fictitious forces

When a rotating reference frame is used as reference, an inertial (also known as fictitious or pseudo) centrifugal force appears, in order to make Newton's laws of motion valid in such a frame. That is, the true force on a mass must be supplemented by a (fictitious) centrifugal force that is directed away from the axis of rotation, and also a Coriolis force that bends a moving object's path.

Returning to the example of the car turning, if we consider the reference frame that is rotating together with the car (a model which those inside the car often will find natural), it feels as if a 'magical' force is pushing the passenger away from the center of the bend. In the rotating frame of the car, this push is ascribed to a fictitious force (as opposed to an actual force exerted by another object), an inertial force called centrifugal force. The centrifugal force is invoked by the passengers because the car's acceleration is hidden from observers in the reference frame moving with the car. Nevertheless, in the rotating frame, this type of force appears as real and a natural basis for analysis. [For more detail see [http://www.spacefuture.com/archive/artificial_gravity_and_the_architecture_of_orbital_habitats.shtml Hall: "Artificial gravity and the architecture of orbital habitats"] .] [ [http://www.spacefuture.com/archive/inhabiting_artificial_gravity.shtml Hall: "Inhabiting artificial gravity] ] . See the articles on centrifugal force and fictitious force for examples.Fictitious forces do not appear, however, in inertial frames of reference, in particular, when rotating frames of reference are mapped to inertial frames for the physics.cite book |author= M. Alonso & E.J. Finn |title=Fundamental university physics
publisher=, Addison-Wesley |year=1992 |url=http://books.google.com/books?id=c5UAAAAACAAJ&dq=isbn:0201565188&lr=&as_brr=0 |isbn= 0201565188] .

See also

* Circular motion
* Coriolis force
* Centripetal force
* Rotational motion
* Centrifugal force

References

External links

* [http://www.infoplease.com/ce6/sci/A0811114.html Centrifugal reaction force] - Columbia electronic encyclopedia
* [http://www.bbc.co.uk/dna/h2g2/A597152 Centrifugal Force] at h2g2
* [http://regentsprep.org/Regents/physics/phys06/bcentrif/default.htm Centripetal force] vs. [http://regentsprep.org/Regents/physics/phys06/bcentrif/centrif.htm Centrifugal force] - from an online Regents Exam physics tutorial by the Oswego City School District
* [http://www.npl.washington.edu/av/altvw55.html Centrifugal force acts inwards near a black hole]
* [http://www.mathpages.com/home/kmath633/kmath633.htm Centripetal and Centrifugal Forces] at MathPages