Centrifugal force (from Latin centrum "center" and fugere "to flee") is a term which may refer to two different forces which are related to rotation. Both of them are oriented away from the axis of rotation, but the object on which they are exerted differs.

• The reactive centrifugal force is the reaction to the centripetal force ^*. This is equal in magnitude to the centripetal force, directed away from the center of rotation, and is exerted by the rotating object upon the object which exerts the centripetal force. As it is an actual force, it is always present, independent of the choice of reference frame. Note that this Newtonian use of the term centrifugal force is rare in modern discussions.

• The fictitious centrifugal force appears when a rotating reference frame is used for analyzing the system. The centrifugal force is exerted on all objects, and directed away from the axis of rotation.

Both of the above can be observed in action on a passenger riding in a car. If the car swerves around a corner, the passenger's body pushes against the outer edge of the car. This is the reactive centrifugal force, which is called a reaction force because it results from passive interaction with the car which actively pushes against the body.

Using a reference frame which is fixed relative to the car (a model which those inside the car will often find natural) and while ignoring its rotation, it looks like an external force is pulling the passenger out of the car. This is the fictitious centrifugal force, so called because it is not an actual force exerted by some other object.

Reactive centrifugal force

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When viewed from an inertial frame of reference, the application of Newton's laws of motion is simple. The passenger's inertia resists acceleration, keeping the passenger moving with constant speed and direction as the car begins to turn. From this point of view, the passenger does not gravitate toward the outside of the car; instead, the car curves to meet the passenger.

Once the car contacts the passenger, it then applies a sidewise force to accelerate him or her around the turn with the car. This force is called a centripetal ("center seeking") force because its vector changes direction to continue to point toward the center of the car's arc as the car traverses it.

If the car is acting upon the passenger, then the passenger must be acting upon the car with an equal and opposite force. Being opposite, this reaction force is directed away from the center, therefore centrifugal. It is critical to realize that this centrifugal force acts upon the car, not the passenger.

The centrifugal reaction force with which the passenger pushes back against the door of the car is trivial and is simply given by:

{| $\backslash mathbf\{F\}\_\backslash mathrm\{centrifugal\}\; \backslash ,$ $=\; -\; m\; \backslash mathbf\{a\}\_\backslash mathrm\{centripetal\}\; \backslash ,$ $=\; m\; \backslash omega^2\; \backslash mathbf\{r\}\_\backslash perp\; \backslash ,$

where $m\backslash ,$ is the mass of the rotating object.

Rotating reference frames

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

{| $\backslash mathbf\{a\}\_\backslash mathrm\{rot\}\backslash ,$ $=\backslash mathbf\{a\}\; -\; 2\backslash mathbf\{\backslash omega\; \backslash times\; v\}\; -\; \backslash mathbf\{\backslash omega\; \backslash times\; (\backslash omega\; \backslash times\; r)\}\; \backslash ,$ $=\backslash mathbf\{a\; +\; a\_\backslash mathrm\{Coriolis\}\; +\; a\_\backslash mathrm\{centrifugal\}\}\; \backslash ,$

where $\backslash mathbf\{a\}\_\backslash mathrm\{rot\}\backslash ,$ is the acceleration relative to the rotating frame, $\backslash mathbf\{a\}\backslash ,$ is the acceleration relative to the inertial frame, $\backslash mathbf\{\backslash omega\}\backslash ,$ is the angular velocity vector describing the rotation of the reference frame, $\backslash mathbf\{v\}\backslash ,$ is the velocity of the body relative to the rotating frame, and $\backslash mathbf\{r\}\backslash ,$ is a vector from an arbitrary point on the rotation axis to the body. A derivation can be found in the article fictitious force.

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

$\backslash mathbf\{a\}\_\backslash textrm\{centrifugal\}\; =\; -\; \backslash mathbf\{\backslash omega\; \backslash times\; (\backslash omega\; \backslash times\; r)\}\; =\; \backslash omega^2\; \backslash mathbf\{r\}\_\backslash perp$

where $\backslash mathbf\{r\_\backslash perp\}$ is the component of $\backslash mathbf\{r\}\backslash ,$ perpendicular to the axis of rotation.

Derivation

If we have two frames, one inertial and one rotating with a constant angular velocity $\backslash vec\; \backslash omega$, a time derivative of a vector in the rotating frame, $\backslash left\; (\; \backslash frac\{d\}\{dt\}\; \backslash right\; )\; \_r$, is transformed to the time derivative in the inertial frame, $\backslash left\; (\; \backslash frac\{d\}\{dt\}\; \backslash right\; )\; \_i$, by the following relation:

$\backslash left\; (\; \backslash frac\{d\}\{dt\}\; \backslash right\; )\; \_i\; =\; \backslash left\; (\; \backslash frac\{d\}\{dt\}\; \backslash right\; )\; \_r\; +\; \backslash vec\; \backslash omega\; \backslash times$

This relationship is one between two operators. Now, acceleration is the second derivative of position with respect to time. So, applying the above transformation to the position vector $\backslash vec\; r$ once gets you:

$\backslash dot\; \backslash vec\; r\_i\; =\; \backslash left\; (\; \backslash frac\{d\; \backslash vec\; r\}\{dt\}\; \backslash right\; )\; \_i\; =\; \backslash left\; (\; \backslash frac\{d\; \backslash vec\; r\}\{dt\}\; \backslash right\; )\; \_r\; +\; \backslash omega\; \backslash times\; \backslash vec\; r$

Putting $\backslash dot\; \backslash vec\; r\_i$ back into the transformation, you get:

$\backslash ddot\; \backslash vec\; r\_i\; =\; \backslash left\; (\; \backslash frac\{d\; \backslash dot\; \backslash vec\; r\}\{dt\}\; \backslash right\; )\; \_i\; =\; \backslash left\; (\; \backslash frac\{d\; \backslash dot\; \backslash vec\; r\}\{dt\}\; \backslash right\; )\; \_r\; +\; \backslash omega\; \backslash times\; \backslash dot\; \backslash vec\; r$

$\backslash ddot\; \backslash vec\; r\_i\; =\; \backslash left\; (\; \backslash frac\{d^2\; \backslash vec\; r\}\{dt^2\}\; \backslash right\; )\; \_i\; =\; \backslash left\; (\; \backslash frac\{d\}\{dt\}\; \backslash right\; )\; \_r\; \backslash left\; (\; \backslash left\; (\; \backslash frac\{d\; \backslash vec\; r\}\{dt\}\; \backslash right\; )\; \_r\; +\; \backslash omega\; \backslash times\; \backslash vec\; r\; \backslash right\; )\; +\; \backslash vec\; \backslash omega\; \backslash times\; \backslash left\; (\; \backslash left\; (\; \backslash frac\{d\; \backslash vec\; r\}\{dt\}\; \backslash right\; )\; \_r\; +\; \backslash omega\; \backslash times\; \backslash vec\; r\; \backslash right\; )$

Because $\backslash vec\; \backslash omega$ is a contant vector - that is the rotating reference frame is rotating constantly in the same direction - its time derivative is zero. So, simplifying:

$\backslash ddot\; \backslash vec\; r\_i\; =\; \backslash left\; (\; \backslash frac\{d^2\; \backslash vec\; r\}\{dt^2\}\; \backslash right\; )\; \_i\; =\; \backslash left\; (\; \backslash frac\{d^2\; \backslash vec\; r\}\{dt^2\}\; \backslash right\; )\; \_r\; +\; \backslash omega\; \backslash times\; \backslash left\; (\; \backslash frac\{d\; \backslash vec\; r\}\{dt\}\; \backslash right\; )\; \_r\; +\; \backslash vec\; \backslash omega\; \backslash times\; \backslash left\; (\; \backslash frac\{d\; \backslash vec\; r\}\{dt\}\; \backslash right\; )\; \_r\; +\; \backslash omega\; \backslash times\; \backslash omega\; \backslash times\; \backslash vec\; r$

$\backslash ddot\; \backslash vec\; r\_i\; =\; \backslash left\; (\; \backslash frac\{d^2\; \backslash vec\; r\}\{dt^2\}\; \backslash right\; )\; \_i\; =\; \backslash left\; (\; \backslash frac\{d^2\; \backslash vec\; r\}\{dt^2\}\; \backslash right\; )\; \_r\; +\; 2\; \backslash vec\; \backslash omega\; \backslash times\; \backslash left\; (\; \backslash frac\{d\; \backslash vec\; r\}\{dt\}\; \backslash right\; )\; \_r\; +\; \backslash omega\; \backslash times\; \backslash omega\; \backslash times\; \backslash vec\; r$

Finally, putting in $\backslash vec\; a$ for $\backslash left\; (\; \backslash frac\{d^2\; \backslash vec\; r\}\{dt^2\}\; \backslash right\; )$ and $\backslash vec\; v\_r$ for $\backslash left\; (\; \backslash frac\{d\; \backslash vec\; r\}\{dt\}\; \backslash right\; )\; \_r$, we get the following:

$\backslash vec\; a\_i\; =\; \backslash vec\; a\_r\; +\; 2\; \backslash vec\; \backslash omega\; \backslash times\; \backslash vec\; v\_r\; +\; \backslash vec\; \backslash omega\; \backslash times\; \backslash left\; (\; \backslash vec\; \backslash omega\; \backslash times\; \backslash vec\; r\; \backslash right\; )$

Moving things to the other side, but reversing one cross-product in each term, you find:

$\backslash vec\; a\_r\; =\; \backslash vec\; a\_i\; +\; 2\; \backslash vec\; v\_r\; \backslash times\; \backslash vec\; \backslash omega\; +\; \backslash vec\; \backslash omega\; \backslash times\; \backslash left\; (\; \backslash vec\; r\; \backslash times\; \backslash vec\; \backslash omega\; \backslash right\; )$

This tells us that $\backslash vec\; a\_r$, the acceleration of some object at $\backslash vec\; r$ as observed by someone at rest in the rotating frame is equal to the acceleration, $\backslash vec\; a\_i$, as observed by an observer in the inertial, non-rotating frame, plus $2\; \backslash vec\; v\_r\; \backslash times\; \backslash vec\; \backslash omega$, which is the coriolis effect's contribution to the acceleration, and $\backslash vec\; \backslash omega\; \backslash times\; \backslash left\; (\; \backslash vec\; r\; \backslash times\; \backslash vec\; \backslash omega\; \backslash right\; )$, which is the centrifugal acceleration term.

Fictitious centrifugal force

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An alternative way of dealing with a rotating frame of reference is to make Newton's laws of motion artificially valid by adding fictitious forces that are pretended 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 fictitious centrifugal force, given by:

{| $\backslash mathbf\{F\}\_\backslash mathrm\{centrifugal\}\; \backslash ,$ $=\; m\; \backslash mathbf\{a\}\_\backslash mathrm\{centrifugal\}\; \backslash ,$ $=m\; \backslash omega^2\; \backslash mathbf\{r\}\_\backslash perp\; \backslash ,$

where $m\backslash ,$ is the mass of the object.

This 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 fictitious Coriolis force. 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 $-m\; \backslash omega^2\; \backslash mathbf\{r\}\_\backslash perp$ required to account for this apparent rotation is the sum of the centrifugal force ($m\; \backslash omega^2\; \backslash mathbf\{r\}\_\backslash perp$) and the Coriolis force ($-2m\; \backslash mathbf\{\backslash omega\; \backslash times\; v\}\; =\; -2m\; \backslash omega^2\; \backslash mathbf\{r\}\_\backslash perp$). Since this centripetal force includes contributions from only fictitious forces, it has no reactive counterpart.

Potential energy of the fictitious centrifugal force

The fictitious centrifugal force can be described by a potential energy of the form

$E\_p\; =\; -\backslash frac\{1\}\{2\}\; m\; \backslash omega^2\; r\_\backslash perp^2$

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

Similarly, the potential energy of the centrifugal force is often used in the calculation of the height of the tides 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 centrifuge 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.

Centrifugal forces in statics

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Consider a ball that swings around a stationary pivot to which it is tethered by a light, strong rope. There is tension in the rope, pulling inwards on the ball (the centripetal force) and simultaneously pulling outwards on the pivot (the reactive centrifugal force). The tension is real, so these two forces still exist if we move to a corotating frame. However, in the rotating frame there is also a fictitious centrifugal force that pulls outwards on the ball. It is distinct from the reactive centrifugal force that pulls outward on the pivot.

When solving statics problems in a rotating frame (e.g. when calculating the internal stresses in a flywheel) it is convenient to think of the fictitious centrifugal force as being transmitted through the rope and becoming the pull on the pivot. In statics one often considers a force "the same" before and after it has been conveyed by a structural element, so according to this view the reaction force on the pivot is the fictitious force.

This identification often leads to confusion about the "fictitious" nature of the centrifugal force, because the pull on the pivot is a perfectly real force. The confusion can be resolved by noting that the distinction between fictitious and real forces depends on the frame of reference that one chooses for the laws of physics. On the other hand, considering the reaction force to be the fictitious force is only valid in statics, that is, once we have decided to always use that particular reference frame in which the entire system is stationary. The convenience of viewing a transmitted force as the same as the original force comes at the cost of a meaningful distinction between whether a force is real or fictitious.

Confusion and misconceptions

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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 (which is true for physics in general). 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 fictitious forces entirely by analyzing systems using inertial frames of reference 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, science teachers often de-emphasize the fictitious centrifugal force that appears to exist in a rotating reference frame. However, in their zeal to stamp out the misunderstanding of the term in this one case, they may try to expunge it from the language entirely.

Applications

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• A Centrifugal governor 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 throttle.
• Centrifugal forces can be used to generate artificial gravity. Proposals have been made to have gravity generated in space stations designed to rotate. The Mars Gravity Biosatellite will use study the effects of Mars level gravity on mice with simulated gravity from centrifugal force.
• Centrifuges are used in science and industry to separate substances by their relative masses.
• Some Amusement park rides makes 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.

See also

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• Centripetal force
• Circular motion
• Coriolis force

References and external links

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• Newton's description in Principia
• Centrifugal force - Columbia electronic encyclopedia
• Centrifugal Force - from ScienceWorld
• Java applet demonstrating centrifugal and Coriolis forces
• M. Alonso and E.J. Finn, Fundamental university physics, Addison-Wesley

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