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#REDIRECT ]
{{expert-subject|Physics}}
{{Mergefrom|Reactive centrifugal force|date=May 2008}}

:''For the real outward-acting force that can be found in circular motion see ]''
In ], '''centrifugal force''' (from ] ''centrum'' "center" and ''fugere'' "to flee") is a ]<ref>{{cite web
|url=http://scienceworld.wolfram.com/physics/CentrifugalForce.html
|title=Centrifugal Force
}}</ref><ref>{{cite web
|url=http://www.britannica.com/eb/article-9022100/centrifugal-force
|title=Centrifugal Force - Britannica online encyclopedia
}}</ref> that appears when describing physics in a ] and applies to anything with ] considered in such a frame.

Changing coordinates from an ] to a rotating one alters the equations of motion to include ] to compensate for the non-inertial character of the frame {{Harv|Marion|Thornton|1995|loc=Ch.10}}. These fictitious forces have two components: one depends only on the position and the mass of the object and is always oriented away from the ] of the rotating frame; this is the centrifugal force; the other is the ] and depends on the velocity and mass of the object but is independent of its position {{Harv|Marion|Thornton|1995|p=386}}.

In certain situations a rotating reference frame has advantages over an inertial reference frame {{Harv|Marion|Thornton|1995|p=387}}. For example, it is much more convenient to describe what happens on the inside of a car going around a corner or in a centrifuge from a co-rotating frame.

== Is centrifugal force a "real" force? ==

In the rotating frame of reference the centrifugal and Coriolis forces appear to be real physical forces, but both these forces are called "''fictitious''" because the effects ascribed to them in the rotating frame can be described equally well in an inertial frame ''without'' them (moreover, these "forces" do not obey ]: they have no associated reaction force). From the point of view of an observer in an inertial frame, the centrifugal and Coriolis forces can have real physical effects in situations where the object in question is co-rotating such is in the case of the centrifuge device. In situations in which the object in question is not co-rotating, these fictitious forces are merely ] of ]. The distinction between these two aspects of fictitious forces is the subject of a long standing debate known as the ].Classifying such forces as "''fictitious''" reflects the special role of ] in ]. Still, to those who actually live in a ''non-inertial'' frame such as the rotating planet Earth, fictitious forces are a very real part of everyday experience. They also provide a simple way of discussing forces and motions within rotating environments such as centrifuges, carousels, turning cars, and spinning buckets.

== Rotating reference frames==
Rotating reference frames are sometimes used in physics, mechanics, or ] where they are the most convenient frame to use.

The laws of physics are the same in all ], but not in ]s. When using a rotating reference frame, the laws of physics are mapped from the most convenient inertial frame to the 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 (see '']'' for a derivation) {{Harv|Marion|Thornton|1995|pp=386–387}}

:{|
|-
|<math>\mathbf{a}_\mathrm{rot}\,</math>
|<math>=\mathbf{a} - 2\mathbf{\Omega \times v_\mathrm{rot}} - \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 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,<ref>This vector points along the axis of rotation with polarity determined by the ] and a magnitude |Ω| = ω = angular rate of rotation.</ref> <math>\mathbf{v_\mathrm{rot}}\,</math> is the velocity of the body relative to the rotating frame, and <math>\mathbf{r}\,</math> is the position vector of the body.
The last term is the centrifugal acceleration, so:

:<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 \, \boldsymbol{\Omega} \times \boldsymbol 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 \boldsymbol{\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.

==Applications==
The operations of numerous common rotating mechanical systems are most easily conceptualized in terms of centrifugal force. For example:
* A ] regulates the speed of an engine by using spinning masses that move radially, adjusting the ], as the engine changes speed. In the reference frame of the spinning masses, centrifugal force causes the radial movement.
* 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 ], as in proposed designs for rotating space stations. The ] will study the effects of ]-level gravity on mice with gravity simulated in this way.
* ]s 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 ]s 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 ] as generated by centrifugal force as opposed to being generated by gravity.
*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.

Nevertheless, all of these systems can also be described 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.

== See also ==
{{Wiktionary|centrifugal force}}
{{Wiktionary|centrifugal}}
* ]
* ]
* ]
* ] - a force that appears when the frame angular rotation rate varies
* ]
* ] - a force that occurs as ] due to a centripetal force

==Notes==
<references/>

==References==
* {{Citation| last1=Marion |first1=J.B. |last2=Thornton |first2=S.T. |title=Classical Dynamics of Particles and Systems |edition=4th |year=1995 |publisher=Saunders College Publishing}}

*
* - Columbia electronic encyclopedia
* 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
*
* at the HyperPhysics concepts site

==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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Latest revision as of 12:14, 31 July 2015

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