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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 appears when describing physics in a rotating reference frame and applies to anything with mass considered in such a frame.

Changing coordinates from an inertial frame of reference to a rotating one alters the equations of motion to include fictitious forces to compensate for the non-inertial character of the frame (Marion & Thornton 1995, 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 axis of rotation of the rotating frame; this is the centrifugal force; the other is the Coriolis force and depends on the velocity and mass of the object but is independent of its position (Marion & Thornton 1995, p. 386).

In certain situations a rotating reference frame has advantages over an inertial reference frame (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 Newton's third law: 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 artifacts of coordinate transformation. The distinction between these two aspects of fictitious forces is the subject of a long standing debate known as the Bucket argument.Classifying such forces as "fictitious" reflects the special role of inertial frames in Newtonian mechanics. 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 meteorology where they are the most convenient frame to use.

The laws of physics are the same in all inertial frames, but not in rotating reference frames. 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 Fictitious force for a derivation) (Marion & Thornton 1995, pp. 386–387)

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

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, ${\displaystyle \mathbf {v_{\mathrm {rot} }} \,}$ is the velocity of the body relative to the rotating frame, and ${\displaystyle \mathbf {r} \,}$ is the position vector of the body. The last term is the centrifugal acceleration, so:

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

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

Notes

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.

References

• Marion, J.B.; Thornton, S.T. (1995), Classical Dynamics of Particles and Systems (4th ed.), Saunders College Publishing

• 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