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

In physics, the centrifugal force (from Latin centrum "center" and fugere "to flee") is an inertial force which is apparent in rotating reference frames and applies to every object or fluid element under consideration. This force is oriented away from the axis of rotation of the reference frame.

To the extent to which the object or fluid element co-rotates with the frame, a radial acceleration or a hydrostatic pressure is induced.

One practical application of centrifugal force is the centrifuge device which causes heavier particles in a solution to drift towards the edge.

The term centrifugal force should not be confused with the term centripetal force. The lattter is a term, generally restricted to circular motion, and it describes an inward acting force which forms an action-reaction pair with the centrifugal force.

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 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 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.

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

where ${\displaystyle \mathbf {a} _{\mathrm {rot} }\,}$ is the frame 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. A derivation can be found in the article fictitious force.

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":

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

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 (${\displaystyle -2m\mathbf {\omega \times v} =-2m\omega ^{2}\mathbf {r} _{\perp }}$). 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.

Confusion and misconceptions

One can often avoid dealing with pseudo 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.

Although centrifugal force is often described as 'fictitious', it isn't imaginary in the sense that Sherlock Holmes is fiction with no relation to reality. Centrifugal force gives physically very real effects, ultimately the centrifugal and Coriolis effects are behaviours which are due to inertia. Some writers, notably Richard Feynman preferred to use the term 'pseudo force' which has connotations of pretense rather than outright fiction.

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. 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 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.
• 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. Proposals have been made to have gravity generated in space stations designed to rotate. The Mars Gravity Biosatellite will 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 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.

See also

• Circular motion
• Coriolis force
• Centripetal force
• Rotational motion
• Reactive centrifugal force - a force that occurs as reaction due to a centripetal force

References

• Newton's description in Principia
• Centrifugal reaction force - Columbia electronic encyclopedia
• Fictitious centrifugal force - from ScienceWorld
• 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

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
• XKCD demonstrates the life and death importance of centrifugal force

• Articles with minor POV problems from December 2007
• Force
• Mechanics