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:''For the real outward-acting force that exists in all circular motion see ]''
In ], the '''] centrifugal force''' (from ] ''centrum'' "center" and ''fugere'' "to flee") is a ] which is apparent in ]s and applies to every object under consideration. This force is oriented away from the ] of the reference frame.

In some cases, it is convenient to use a rotating reference frame, rather than an inertial reference frame. When this is desirable, coordinate transformations from the inertial reference frame can be applied.

However, to do this correctly, in the rotating reference frame, it turns out that a centrifugal force must be applied in conjunction with a ] for the correct equation of motion to be calculated. The centrifugal force depends only on the position and mass of the object it applies to (and does not depend on its velocity), whereas the Coriolis force depends on the velocity and mass of the object but is independent of its position.

The term 'centrifugal force' is often misused in everyday discussions; the centrifugal forces discussed in this article only appear in rotating reference frames.

== Rotating reference frames==
Rotating reference frames are sometimes used in physics, mechanics or ] where they are the most convenient frame to use. For example the surface of the Earth is only stationary in a reference frame that rotates once per day. For many purposes the rotation causes negligible effects, but for some phenomena such as weather systems this rotation cannot be ignored.

In the classical approach, the inertial frame remains the true reference for the laws of mechanics and analysis. When using a ], the laws of physics are coordinate mapped from the most convenient inertial frame to a rotating frame. Assuming a constant rotation speed, this is achieved by adding to every object two ''coordinate accelerations'' which correct for the constant rotation of the coordinate axes.

:{|
|-
|<math>\mathbf{a}_\mathrm{rot}\,</math>
|<math>=\mathbf{a} - 2\mathbf{\omega \times v} - \mathbf{\omega \times (\omega \times r)} \,</math>
|-
|
|<math>=\mathbf{a + a_\mathrm{Coriolis} + a_\mathrm{centrifugal}} \,</math>
|}

where <math>\mathbf{a}_\mathrm{rot}\,</math> is the frame acceleration relative to the rotating frame, <math>\mathbf{a}\,</math> is the acceleration relative to the inertial frame, <math>\mathbf{\omega}\,</math> is the ] vector describing the rotation of the reference frame, <math>\mathbf{v}\,</math> is the velocity of the body relative to the rotating frame, and <math>\mathbf{r}\,</math> is the position vector of a point on the body. A derivation can be found in the article ].

The last term is the centrifugal acceleration, so we have:

:<math> \mathbf{a}_\textrm{centrifugal} = - \mathbf{\omega \times (\omega \times r)} = \omega^2 \mathbf{r}_\perp </math>

where <math>\mathbf{r_\perp}</math> is the component of <math>\mathbf{r}\,</math> perpendicular to the axis of rotation.

== Fictitious forces ==
{{main|Fictitious force}}
An alternative way of dealing with a rotating frame of reference is to make Newton's laws of motion artificially valid by adding pseudo forces to be the cause of the above acceleration terms. In particular, the centrifugal acceleration is added to the motion of every object, and attributed to a centrifugal force, given by:

:{|
|-
|<math>\mathbf{F}_\mathrm{centrifugal} \,</math>
|<math>= m \mathbf{a}_\mathrm{centrifugal} \,</math>
|-
|
|<math>=m \omega^2 \mathbf{r}_\perp \,</math>
|}

where <math>m\,</math> is the mass of the object.

This pseudo or fictitious centrifugal force is a sufficient correction to Newton's second law only if the body is stationary in the rotating frame. For bodies that move with respect to the rotating frame it must be supplemented with a second pseudo force, the "]":

:<math>\mathbf{F}_\mathrm{coriolis} = -2 \, m \, \vec \omega \times \vec v</math>

For example, a body that is stationary relative to the ''non''-rotating frame, will be rotating when viewed from the rotating frame. The ''centripetal'' force of <math>-m \omega^2 \mathbf{r}_\perp</math> required to account for this apparent rotation is the sum of the centrifugal pseudo force (<math>m \omega^2 \mathbf{r}_\perp</math>) and the Coriolis force
(<math>-2m \mathbf{\omega \times v} = -2m \omega^2 \mathbf{r}_\perp</math>). Since this centripetal force includes contributions from only pseudo forces, it has no reactive counterpart.

== Potential energy ==
] liquids rotating around a vertical axis is an upward-opening circular paraboloid.]]

The fictitious centrifugal force is ] and has a ] of the form

:<math>E_p = -\frac{1}{2} m \omega^2 r_\perp^2</math>

This is useful, for example, in calculating the form of the water surface <math>h(r)\,</math> in a rotating bucket: requiring the potential energy per unit mass on the surface <math>gh(r) - \frac{1}{2}\omega^2 r^2\,</math> to be constant, we obtain the ] form <math>h(r) = \frac{\omega^2}{2g}r^2 + C</math> (where <math>C</math> is a constant).

Similarly, the potential energy of the centrifugal force is often used in the calculation of the height of the ]s on the Earth (where the centrifugal force is included to account for the rotation of the Earth around the Earth-Moon center of mass).

The principle of operation of the ] also can be simply understood in terms of this expression for the potential energy, which shows that it is favorable energetically when the volume far from the axis of rotation is occupied by the heavier substance.

The coriolis force has no equivalent potential, as it acts perpendicular to the velocity vector and hence rotates the direction of motion, but does not change the energy of a body.

==Real versus fictional forces==
{| class="wikitable" align="right"
|
! align=center| Real force
! align=center| Pseudo force
|-
! align=center| Reference<br>frame
| align=center| Any
| align=center| Any rotating system
|-
! align=center| Exerted by
| align=center| Bodies moving along<br>curved paths
| align=center| Acts as if exerted by the rotation axis<br>of the frame of reference
|-
! align=center| Exerted upon
| align=center| The object imposing<br>curved motion
| align=center| All bodies
|-
! align=center| Direction
| align=center| Away from the<br>]
| align=center| Away from the rotation axis<br>of the frame of reference
|}

*A mass undergoing ] constantly ]s toward the center of the circle. This ] is caused by a ], which is applied to 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 "''']'''" centrifugal force: it is directed away from the ], and is exerted ''by'' the rotating mass ''on'' the object which imposes the centripetal acceleration. Although this sense was used by ],<ref></ref> it is only occasionally used in modern discussions.<ref></ref><ref></ref><ref></ref><ref>http://physnet.org/modules/pdf_modules/m17.pdf
ACCELERATION AND FORCE IN CIRCULAR MOTION by Peter Signell</ref>

* An '''inertial''' (also known as ''']''' or '''pseudo''') centrifugal force appears when a rotating ] is used for analysis. To make ] valid in such a frame, the true force on a mass must be supplemented by a (fictitious) centrifugal force that is directed away from the axis of rotation.

Both of the above can be easily observed in action for a passenger riding in a car. If a car swerves around a corner, a passenger's body seems to move towards the outer edge of the car and then pushes against the door.

In the reference frame that is rotating together with the car (a model which those inside the car will often find natural), it looks as if a force is pushing the passenger away from the center of the bend. This is a ]&mdash;not an actual force exerted by any other object. The effect occurs when the reference frame is the car, because that ignores the car's ]. Physicists sometimes treat this type of force much as if it were a real force, as it makes calculations simpler and gives correct results.

However, the force with which the passenger pushes against the door is real. That force is called a ''reaction force'' because it results from passive interaction with the car which actively pushes against the body. As it is directed outward, it is a centrifugal force. Note that this ''real'' ] does not appear until the person touches the body of the car (ignoring any force exerted by the seat on the person's body, etc.). The car also exerts an equal but ] on the person, called a "]".

==Confusion and misconceptions==
Centrifugal force can be a confusing term because it is used (or misused) in more than one instance, and because sloppy labelling 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).

One can avoid dealing with pseudo forces entirely by analyzing systems using ] for the physics; and when convenient, one simply maps to a rotating frame without forgetting about the frame rotation, as shown above. Such is standard practice in mechanics textbooks.

Because rotating frames are not vital for understanding mechanics, they are often not discussed in science education. Therefore teachers who need to impress on their students that centrifugal forces have no place in their calculations often do not have occasion to give a matching emphasis to the fact that a centrifugal force does occur in a rotating frame. As a result, even students who master the physics curriculum may leave school with the false impression that it is ''never'' scientifically valid to speak about centrifugal forces.{{POV-statement|date=December 2007}} Nevertheless, many popular discussions of forces do use the term "centrifugal", without pointing out that it is fictitious, and assume the reader is knowledgeable of the true inertial character of the force, leading to misconceptions and bad use of the term.

==Applications==
* A ] regulates the speed of an engine by using spinning masses that respond to centrifugal force generated by the engine. If the engine increases in speed, the masses move and trigger a cut in the ].
* A ] 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.
* Centrifugal forces can be used to generate ]. Proposals have been made to have gravity generated in space stations designed to rotate. The ] will study the effects of ] level gravity on mice with simulated gravity from centrifugal force.
* ]s are used in science and industry to separate substances by their relative masses.
*Some ] ]s make use of centrifugal forces. For instance, a ]’s spin forces riders against a wall and allows riders to be elevated above the machine’s floor in defiance of Earth’s gravity.
*] and ] are production methods that uses centrifugal force to disperse liquid metal or plastic throughout the negative space of a mold.

== See also ==
{{Wiktionary|centrifugal}}
* ]
* ]
* ]
* ]

==References==

<references/>
*
* - Columbia electronic encyclopedia
* - from ]
* M. Alonso and E.J. Finn, ''Fundamental university physics'', Addison-Wesley
* vs. - from an online Regents Exam physics tutorial by the Oswego City School District
*

==External links==
* showing scenes as viewed from both an inertial frame and a rotating frame of reference, visualizing the Coriolis and centrifugal forces.
* at MathPages
* at h2g2

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Revision as of 12:13, 28 April 2008

Centrifugal force (rotating reference frame): Difference between revisions Add topic