Centrifugal force
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Classical mechanics  

<math>\vec{F} = m\vec{a}</math>  
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</table> In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) directed away from the axis of rotation that appears to act on all objects when viewed in a rotating frame of reference. The concept of the centrifugal force can be applied in rotating devices, such as centrifuges, centrifugal pumps, centrifugal governors, and centrifugal clutches, and in centrifugal railways, planetary orbits and banked curves, when they are analyzed in a rotating coordinate system. The term has sometimes also been used for the reactive centrifugal force that is a reaction to a centripetal force. ContentsIntroductionThe centrifugal force is an outward force apparent in a rotating reference frame; it does not exist when a system is described relative to an inertial frame of reference.^{[1]} All measurements of position and velocity must be made relative to some frame of reference. For example, an analysis of the motion of an object in an airliner in flight could be made relative to the airliner, to the surface of the Earth, or even to the Sun.^{[2]} A reference frame that is at rest (or one that moves with no rotation and at constant velocity) relative to the "fixed stars" is generally taken to be an inertial frame. Any system can be analyzed in an inertial frame (and so with no centrifugal force). However, it is often more convenient to describe a rotating system by using a rotating framethe calculations are simpler, and descriptions more intuitive. When this choice is made, fictitious forces, including the centrifugal force, arise. In a rotating reference frame, all objects, regardless of their state of motion, appear to be under the influence of a radially (from the axis of rotation) outward force that is proportional to their mass, to the distance from the axis of rotation of the frame, and to the square of the angular velocity of the frame.^{[3]}^{[4]} This is the centrifugal force. Motion relative to a rotating frame results in another fictitious force: the Coriolis force. If the rate of rotation of the frame changes, a third fictitious force (the Euler force) is required. These fictitious forces are necessary for the formulation of correct equations of motion in a rotating reference frame^{[5]}^{[6]} and allow Newton's laws to be used in their normal form in such a frame (with one exception: the fictitious forces do not obey Newton's third law: they have no equal and opposite counterparts).^{[5]} ExamplesTurning vehicleA common experience that gives rise to the idea of a centrifugal force is encountered by passengers riding in a vehicle, such as a car, that is changing direction. If a car is traveling at a constant speed along a straight road, then a passenger inside is not accelerating and, according to Newton's second law of motion, the net force acting on her is therefore zero (all forces acting on her cancel each other out). If the car enters a curve that bends to the left, the passenger experiences an apparent force that seems to be pulling her towards the right. This is the fictitious centrifugal force. It is needed within the passenger's local frame of reference to explain her sudden tendency to start accelerating to the right relative to the car—a tendency which she must resist by applying a rightward force to the car (for instance, a frictional force against the seat) in order remain in a fixed position inside. Since she pushes the seat toward the right, Newton's third law says that the seat pushes her toward the left. The centrifugal force must be included in the passenger's reference frame (in which the passenger remains at rest): it counteracts the leftward force applied to the passenger by the seat, and explains why this otherwise unbalanced force does not cause her to accelerate.^{[7]} However, it would be apparent to a stationary observer watching from an overpass above that the frictional force exerted on the passenger by the seat is not being balanced; it constitutes a net force to the left, causing the passenger to accelerate toward the inside of the curve, as she must in order to keep moving with the car rather than proceeding in a straight line as she otherwise would. Thus the "centrifugal force" she feels is the result of a "centrifugal tendency" caused by inertia.^{[8]} Similar effects are encountered in planes and roller coasters where the magnitude of the apparent force is often reported in "G's". A stone on a stringIf a stone is whirled round on a string, in a horizontal plane, the only real force acting on the stone in the horizontal plane is applied by the string (gravity acts vertically). No other force acts on the stone so there is a net force on the stone in the horizontal plane. In an inertial frame of reference, were it not for this net force acting on the stone, the stone would travel in a straight line, according to Newton's first law of motion. In order to keep the stone moving in a circular path, a centripetal force, in this case provided by the string, must be continuously applied to the stone. As soon as it is removed (for example if the string breaks) the stone moves in a straight line. In this inertial frame, the concept of centrifugal force is not required as all motion can be properly described using only real forces and Newton's laws of motion. In a frame of reference rotating with the stone around the same axis as the stone, the stone is stationary. However, the force applied by the string is still acting on the stone. If one were to apply Newton's laws in their usual (inertial frame) form, one would conclude that the stone should accelerate in the direction of the net applied forcetowards the axis of rotationwhich it does not do. The centrifugal force and other fictitious forces must be included along with the real forces in order to apply Newton's laws of motion in the rotating frame. EarthThe Earth, because it rotates once a day on its axis, constitutes a rotating reference frame. Because the rotation is slow, the fictitious forces it produces are small, and in everyday situations can generally be neglected. Even in calculations requiring high precision, the centrifugal force is generally not explicitly included, but rather lumped in with the gravitational force: the strength and direction of the local "gravity" at any point on the Earth's surface is actually a combination of gravitational and centrifugal forces. Weight of an object at the poles and on the equatorIf an object is weighed with a simple spring balance at one of the Earth's poles, there are two forces acting on the object: the Earth's gravity, which acts in a downward direction, and the equal and opposite tension in the spring, acting upward. There is no net force acting on the object and the spring balance so the object does not accelerate and remains stationary. The balance shows the value of the force of gravity on the object. When the same object is weighed on the equator the same two real forces act upon the object. However, the object is moving in a circular path as the Earth rotates. When considered in an inertial frame (that is to say, one that is not rotating with the Earth), some of the force of gravity is expended just to keep the object in its circular path (centripetal force). Less tension in the spring is required to counteract the 'remaining' force of gravity. Less tension in the spring would be reflected on a scale as less weight — about 0.3% less at the equator than at the poles.^{[9]} In the Earth reference frame (in which the object being weighed is at rest), this difference is explained by the centrifugal force. Note: In fact, the observed weight difference is more — about 0.53%. Earth's gravity is a bit stronger at the poles than at the equator, because the Earth is not a perfect sphere, so an object at the poles is slightly closer to the center of the Earth than one at the equator; this effect combines with the centrifugal force to produce the observed weight difference.^{[10]} An equatorial railwayThis thought experiment is more complicated than the previous examples in that it requires the use of the Coriolis force as well as the centrifugal force. If there were a railway line running round the Earth's equator, a train moving westward along it fast enough would remain stationary in a frame moving (but not rotating) with the Earth; it would stand still as the Earth spun beneath it. In this inertial frame the situation is easy to analyze. The only forces acting on the train (assuming no wind resistance or other horizontal forces) are its gravity (downward) and the equal and opposite (upward) force from the track. There is no net force on the train and it therefore remains stationary. In a frame rotating with the Earth the train moves in a circular orbit as it travels round the Earth. In this frame, the upward reaction force from the track and the force of gravity on the train remain the same, as they are real forces. However, in the Earth's (rotating) frame, the train is traveling in a circular path and therefore requires a centripetal (downward) force to keep it on this path. Because this uses a rotating frame, the (fictitious) centrifugal force must be applied to the train. This is equal in value to the required centripetal force but acts in an upward direction — the opposite direction to that required. It would seem that there is a net upward force on the train and it should therefore accelerate upward. The resolution to this paradox lies in the fact that the train is in motion with respect to the rotating frame and is subject to (in addition to the centrifugal force) the Coriolis force, which, in this example, acts downward direction and is twice as strong as centrifugal force. DerivationMain article: Rotating reference frame
Main articles: Fictitious force and Mechanics of planar particle motion
For the following formalism, the rotating frame of reference is regarded as a special case of a noninertial reference frame that is rotating relative to an inertial reference frame denoted the stationary frame. Time derivatives in a rotating frameIn a rotating frame of reference, the time derivatives of any vector function P of time—such as the velocity and acceleration vectors of an object—will differ from its time derivatives in the stationary frame. If P_{1} P_{2}, P_{3} are the components of P with respect to unit vectors i, j, k directed along the axes of the rotating frame,^{[11]} then the first time derivative [dP/dt] of P with respect to the rotating frame is, by definition, dP_{1}/dt i + dP_{2}/dt j + dP_{3}/dt k. If the absolute angular velocity of the rotating frame is ω then the derivative dP/dt of P with respect to the stationary frame is related to [dP/dt] by the equation:^{[12]} <math>\frac{\operatorname{d}\boldsymbol{P {\operatorname{d}t} = \left[\frac{\operatorname{d}\boldsymbol{P}}{\operatorname{d}t}\right] + \boldsymbol{\omega} \times \boldsymbol{P}\ ,</math>}}
where <math>\times</math> denotes the vector cross product. In other words, the rate of change of P in the stationary frame is the sum of its apparent rate of change in the rotating frame and a rate of rotation <math>\boldsymbol{\omega} \times \boldsymbol{P}</math> attributable to the motion of the rotating frame. The vector ω has magnitude ω equal to the rate of rotation and is directed along the axis of rotation according to the righthand rule. AccelerationNewton's law of motion for a particle of mass m written in vector form is:
where F is the vector sum of the physical forces applied to the particle and a is the absolute acceleration (that is, acceleration in an inertial frame) of the particle, given by:
where r is the position vector of the particle. By applying the transformation above from the stationary to the rotating frame three times,^{[13]} the absolute acceleration of the particle can be written as:
\boldsymbol{a} &=\frac{\operatorname{d}^2\boldsymbol{r}}{\operatorname{d}t^2} = \frac{\operatorname{d}}{\operatorname{d}t}\frac{\operatorname{d}\boldsymbol{r}}{\operatorname{d}t} = \frac{\operatorname{d}}{\operatorname{d}t} \left( \left[\frac{\operatorname{d}\boldsymbol{r}}{\operatorname{d}t}\right] + \boldsymbol{\omega} \times \boldsymbol{r}\ \right) \\ &= \left[ \frac{\operatorname{d}^2 \boldsymbol{r}}{\operatorname{d}t^2} \right] + \boldsymbol{\omega}\times \left[ \frac{\operatorname{d} \boldsymbol{r}}{\operatorname{d}t} \right] + \frac{\operatorname{d} \boldsymbol{\omega}}{\operatorname{d}t}\times\boldsymbol{r} + \boldsymbol{\omega} \times \frac{\operatorname{d}\boldsymbol{r}}{\operatorname{d}t} \\ &= \left[ \frac{\operatorname{d}^2 \boldsymbol{r}}{\operatorname{d}t^2} \right] + \boldsymbol{\omega}\times \left[ \frac{\operatorname{d} \boldsymbol{r}}{\operatorname{d}t} \right] + \frac{\operatorname{d} \boldsymbol{\omega}}{\operatorname{d}t}\times\boldsymbol{r} + \boldsymbol{\omega} \times \left( \left[\frac{\operatorname{d}\boldsymbol{r}}{\operatorname{d}t}\right] + \boldsymbol{\omega} \times \boldsymbol{r}\ \right) \\ &= \left[ \frac{\operatorname{d}^2 \boldsymbol{r}}{\operatorname{d}t^2} \right] + \frac{\operatorname{d} \boldsymbol{\omega}}{\operatorname{d}t}\times\boldsymbol{r} + 2 \boldsymbol{\omega}\times \left[ \frac{\operatorname{d} \boldsymbol{r}}{\operatorname{d}t} \right] + \boldsymbol{\omega}\times ( \boldsymbol{\omega} \times \boldsymbol{r}) \ . \end{align} </math> ForceThe apparent acceleration in the rotating frame is [d^{2}r/dt^{2}]. An observer unaware of the rotation would expect this to be zero in the absence of outside forces. However, Newton's laws of motion apply only in the inertial frame and describe dynamics in terms of the absolute acceleration d^{2}r/dt^{2}. Therefore, the observer perceives the extra terms as contributions due to fictitious forces. These terms in the apparent acceleration are independent of mass; so it appears that each of these fictitious forces, like gravity, pulls on an object in proportion to its mass. When these forces are added, the equation of motion has the form:^{[14]}^{[15]}^{[16]}
From the perspective of the rotating frame, the additional force terms are experienced just like the real external forces and contribute to the apparent acceleration.^{[17]}^{[18]} The additional terms on the force side of the equation can be recognized as, reading from left to right, the Euler force <math>m \operatorname{d}\boldsymbol{\omega}/\operatorname{d}t \times\boldsymbol{r}</math>, the Coriolis force <math>2m \boldsymbol{\omega}\times \left[ \operatorname{d} \boldsymbol{r}/\operatorname{d}t \right]</math>, and the centrifugal force <math>m\boldsymbol{\omega}\times (\boldsymbol{\omega}\times \boldsymbol{r})</math>, respectively.^{[19]} Unlike the other two fictitious forces, the centrifugal force always points radially outward from the axis of rotation of the rotating frame, with magnitude mω^{2}r, and unlike the Coriolis force in particular, it is independent of the motion of the particle in the rotating frame. As expected, for a nonrotating inertial frame of reference <math>(\boldsymbol\omega=0)</math> the centrifugal force and all other fictitious forces disappear.^{[20]} Absolute rotationMain article: Absolute rotation
Three scenarios were suggested by Newton to answer the question of whether the absolute rotation of a local frame can be detected; that is, if an observer can decide whether an observed object is rotating or if the observer is rotating.^{[21]}^{[22]}
In these scenarios, the effects attributed to centrifugal force are only observed in the local frame (the frame in which the object is stationary) if the object is undergoing absolute rotation relative to an inertial frame. By contrast, in an inertial frame, the observed effects arise as a consequence of the inertia and the known forces without the need to introduce a centrifugal force. Based on this argument, the privileged frame, wherein the laws of physics take on the simplest form, is a stationary frame in which no fictitious forces need to be invoked. Within this view of physics, any other phenomenon that is usually attributed to centrifugal force can be used to identify absolute rotation. For example, the oblateness of a sphere of freely flowing material is often explained in terms of centrifugal force. The oblate spheroid shape reflects, following Clairaut's theorem, the balance between containment by gravitational attraction and dispersal by centrifugal force. That the Earth is itself an oblate spheroid, bulging at the equator where the radial distance and hence the centrifugal force is larger, is taken as one of the evidences for its absolute rotation.^{[23]} ApplicationsThe operations of numerous common rotating mechanical systems are most easily conceptualized in terms of centrifugal force. For example:
Nevertheless, all of these systems can also be described without requiring the concept of centrifugal force, in terms of motions and forces in a stationary frame, at the cost of taking somewhat more care in the consideration of forces and motions within the system. History of conceptions of centrifugal and centripetal forcesMain article: History of centrifugal and centripetal forces
The conception of centrifugal force has evolved since the time of Huygens, Newton, Leibniz, and Hooke who expressed early conceptions of it. Its modern conception as a fictitious force arising in a rotating reference frame evolved in the eighteenth and nineteenth centuries.^{[citation needed]} Centrifugal force has also played a role in debates in classical mechanics about detection of absolute motion. Newton suggested two arguments to answer the question of whether absolute rotation can be detected: the rotating bucket argument, and the rotating spheres argument.^{[25]} According to Newton, in each scenario the centrifugal force would be observed in the object's local frame (the frame where the object is stationary) only if the frame were rotating with respect to absolute space. Nearly two centuries later, Mach's principle was proposed where, instead of absolute rotation, the motion of the distant stars relative to the local inertial frame gives rise through some (hypothetical) physical law to the centrifugal force and other inertia effects. Today's view is based upon the idea of an inertial frame of reference, which privileges observers for which the laws of physics take on their simplest form, and in particular, frames that do not use centrifugal forces in their equations of motion in order to describe motions correctly. The analogy between centrifugal force (sometimes used to create artificial gravity) and gravitational forces led to the equivalence principle of general relativity.^{[26]}^{[27]} Other uses of the termWhile the majority of the scientific literature uses the term centrifugal force to refer to the particular fictitious force that arises in rotating frames, there are a few limited instances in the literature of the term applied to other distinct physical concepts. One of these instances occurs in Lagrangian mechanics. Lagrangian mechanics formulates mechanics in terms of generalized coordinates {q_{k}}, which can be as simple as the usual polar coordinates <math>(r,\ \theta)</math> or a much more extensive list of variables.^{[28]}^{[29]} Within this formulation the motion is described in terms of generalized forces, using in place of Newton's laws the Euler–Lagrange equations. Among the generalized forces, those involving the square of the time derivatives {(dq_{k} ⁄ dt )^{2}} are sometimes called centrifugal forces.^{[30]}^{[31]}^{[32]}^{[33]} In the case of motion in a central potential the Lagrangian centrifugal force has the same form as the fictitious centrifugal force derived in a corotating frame.^{[34]} However, the Lagrangian use of "centrifugal force" in other, more general cases has only a limited connection to the Newtonian definition. In another instance the term refers to the reaction force to a centripetal force, or reactive centrifugal force. A body undergoing curved motion, such as circular motion, is accelerating toward a center at any particular point in time. This centripetal acceleration is provided by a centripetal force, which is exerted on the body in curved motion by some other body. In accordance with Newton's third law of motion, the body in curved motion exerts an equal and opposite force on the other body. This reactive force is exerted by the body in curved motion on the other body that provides the centripetal force and its direction is from that other body toward the body in curved motion.^{[35]}^{[36]} ^{[37]}^{[38]} This reaction force is sometimes described as a centrifugal inertial reaction,^{[39]}^{[40]} that is, a force that is centrifugally directed, which is a reactive force equal and opposite to the centripetal force that is curving the path of the mass. The concept of the reactive centrifugal force is sometimes used in mechanics and engineering. It is sometimes referred to as just centrifugal force rather than as reactive centrifugal force^{[41]}^{[42]} although this usage is deprecated in elementary mechanics.^{[43]} See also
References

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