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It has been suggested that Reactive centrifugal force be merged into this article. (Discuss) Proposed since May 2008.

For the real outward-acting force that can be found in circular motion see Reactive centrifugal force

In physics, centrifugal force (from Latin centrum "center" and fugere "to flee") is a fictitious force that is apparent in a rotating reference frame and applies to anything with mass considered in such a frame. It is oriented away from the axis of rotation of the reference frame.

Sometimes a rotating reference frame has advantages over an inertial reference frame. The best example of this is in the case of the oceans and the atmosphere which co-rotate with the Earth. The centrifugal and Coriolis forces associated with the coordinate transformation, accurately describe the very real physical effects in the oceans and the atmosphere that are induced by this rotation.

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.

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. In an inertial frame, the centrifugal and Coriolis forces are merely artifacts of coordinate transformation. Classifying such forces as "fictitious" reflects the special role of inertial frames in Newtonian mechanics.

Inertial frames have a special role in mechanics, and do not employ fictitious forces. Still, to those who actually live in a non-inertial frame, like Earth-bound humans, 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, weather systems, plughole vortices and spinning buckets.

In former times, centrifugal force was believed to be real. According to Bernoulli, (quote from Bernoulli out of the ET Whittaker book on the history of aethers)

"The elasticity which the Aether appears to possess, and in virtue of which it is able to transmit vibrations, is really due to the presence of these whirlpools; for, owing to centrifugal force, each whirlpool is continually striving to dilate, and so presses against the neighbouring whirlpools."

A quote from Maxwell's 1861 paper 'On Physical Lines of Force' reads,

"The explanation which most readily occurs to the mind is that the excess of pressure in the equatorial direction arises from the centrifugal force of vortices or eddies in the medium having their axes in directions parallel to the lines of force"

One of the strongest arguments in favour of suggesting that centrifugal force is real, lies in the planetary orbital equation. This equation involves an interplay between radially inward gravity and radially outward centrifugal force. If gravity is deemed to be real then it follows that so also must centrifugal force be real since they both act in the radial direction.

Rotating reference frames

Rotating reference frames are sometimes used in physics, mechanics or meteorology where they are the most convenient frame to use. For example the surface of the Earth is stationary only 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.

The laws of physics are the same in all inertial frames. That is not true in a rotating reference frame. When using a rotating reference frame, 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. See fictitious force for a derivation.

${\displaystyle \mathbf {a} _{\mathrm {rot} }\,}$	${\displaystyle =\mathbf {a} -2\mathbf {\Omega \times v} -\mathbf {\Omega \times (\Omega \times r)} \,}$
	${\displaystyle =\mathbf {a+a_{\mathrm {coriolis} }+a_{\mathrm {centrifugal} }} \,}$

where ${\displaystyle \mathbf {a} _{\mathrm {rot} }\,}$ is the acceleration relative to the rotating frame, ${\displaystyle \mathbf {a} \,}$ is the acceleration relative to the inertial frame, ${\displaystyle \mathbf {\Omega } \,}$ is the angular velocity vector describing the rotation of the reference frame, ${\displaystyle \mathbf {v} \,}$ is the velocity of the body relative to the rotating frame, and ${\displaystyle \mathbf {r} \,}$ is the position vector of a point on the body. The last term is the centrifugal acceleration, so we have:

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

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

Fictitious forces

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:

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

where ${\displaystyle m\,}$ 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 "Coriolis force":

${\displaystyle \mathbf {F} _{\mathrm {coriolis} }=-2\,m\,{\boldsymbol {\Omega }}\times {\boldsymbol {v}}}$

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

Potential energy

The fictitious centrifugal force is conservative and has a potential energy of the form

${\displaystyle E_{p}=-{\frac {1}{2}}m\omega ^{2}r_{\perp }^{2}}$

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

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 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.
• 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.
• 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.
• 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.

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

• Circular motion
• Coriolis force
• Centripetal force
• Euler force - a force that appears when the frame angular rotation rate varies
• Rotational motion
• Reactive centrifugal force - a force that occurs as reaction due to a centripetal force

References

1. "Centrifugal Force".
2. "Centrifugal Force - Britannica online encyclopedia".
3. This vector points along the axis of rotation with polarity determined by the right-hand rule and a magnitude |Ω| = ω = angular rate of rotation.

• Newton's description in Principia
• Centrifugal reaction force - Columbia electronic encyclopedia
• M. Alonso and E.J. Finn, Fundamental university physics, Addison-Wesley
• Centripetal force vs. Centrifugal force - from an online Regents Exam physics tutorial by the Oswego City School District
• Centrifugal force acts inwards near a black hole
• Centrifugal force at the HyperPhysics concepts site

External links

• Animation clip showing scenes as viewed from both an inertial frame and a rotating frame of reference, visualizing the Coriolis and centrifugal forces.
• Centripetal and Centrifugal Forces at MathPages
• Centrifugal Force at h2g2

• Articles to be merged from May 2008
• Force
• Mechanics