Definitions

# Centrifugal force (rotating reference frame)

This article is about the fictitious centrifugal force. In classical mechanics, when the motion of an object is described in terms of a reference frame that is rotating about a fixed axis, the expression for the absolute acceleration of the object includes terms involving the rotation rate of the frame. These frame-dependent terms are sometimes brought over to the force side of the equations of motion (with reversed signs), and treated as fictitious forces. "We can apply classical mechanics in noninertial frames if we introduce forces called pseudo-forces (or inertial forces). They are so named because, unlike [real] forces that we have examined so far, we cannot associate them with any particular body in the environment of the particle on which they act."

One of these fictitious forces invariably points directly outward from the axis of rotation, with magnitude proportional to the square of the rotation rate of the frame. In much of the literature on classical dynamics, this term is called centrifugal force.

The apparent motion that may be ascribed to centrifugal force is sometimes called the centrifugal effect.

## Derivation

Newton's law of motion for a particle of mass m can be written in vector form as
$boldsymbol\left\{F\right\} = mboldsymbol\left\{a\right\} ,$
where F is the vector sum of the physical forces applied to the particle and a is the absolute acceleration of the particle, given by
$boldsymbol\left\{a\right\}=frac\left\{d^2\right\}\left\{dt^2\right\}boldsymbol\left\{r\right\} ,$
where r is the position vector of the particle. The differentiations are performed in terms of an inertial reference frame. As shown in Rotating reference frame, for any vector Q depending upon time, its time derivative [dQ/dt] evaluated in terms of a reference frame with a coincident origin but rotating with the absolute angular velocity Ω is related to the absolute derivative dQ/dt by:
$frac\left\{d\right\}\left\{dt\right\}boldsymbol\left\{Q\right\} = left\left[frac\left\{d\right\}\left\{dt\right\}boldsymbol\left\{Q\right\}right\right] + boldsymbol\left\{Omega times Q\right\} ,$
where × denotes the vector cross product and square brackets […] denote evaluation in the rotating frame of reference. As a particular example, it follows that the absolute acceleration of the particle can be written as (for more detail, see Rotating frame of reference):

$boldsymbol\left\{a\right\} = frac\left\{d^2\right\}\left\{dt^2\right\}boldsymbol\left\{r\right\}$

$= left\left[frac\left\{d^2 boldsymbol\left\{r\right\}\right\}\left\{dt^2\right\} right\right] + frac\left\{d boldsymbol\left\{Omega\right\}\right\}\left\{dt\right\}boldsymbol\left\{ times r\right\} + 2 boldsymbol\left\{Omega times\right\} left\left[frac\left\{d boldsymbol\left\{r\right\}\right\}\left\{dt\right\} right\right] + boldsymbol\left\{Omega times\right\} \left(boldsymbol\left\{Omega times r\right\}\right) ,$

It is sometimes convenient to treat the first term on the right hand side as if it actually were the absolute acceleration, and not merely the acceleration in the rotating frame. That is, we pretend the rotating frame is an inertial frame, and move the other terms over to the force side of the equation, and treat them as fictitious forces. When this is done, the equation of motion has the form:

$boldsymbol\left\{F\right\} - mfrac\left\{d boldsymbol\left\{Omega\right\}\right\}\left\{dt\right\} boldsymbol\left\{times r\right\} - 2m boldsymbol\left\{Omega times\right\} left\left[frac\left\{d mathbf\left\{r\right\}\right\}\left\{dt\right\} right\right] - mboldsymbol\left\{Omega times\right\} \left(boldsymbol\left\{Omega times r\right\}\right)$$= mleft\left[frac\left\{d^2 boldsymbol\left\{r\right\}\right\}\left\{dt^2\right\} right\right] .$
The last term on the left hand side (the force side) is commonly called the centrifugal force. It points directly away from the axis of rotation of the rotating reference frame, with magnitude m $\left\{Omega\right\}^2 r$.

A rotating reference frame can have advantages over an inertial reference frame. Sometimes the calculations are simpler (an example is inertial circles), and sometimes the intuitive picture coincides more closely with the rotational frame (an example is sedimentation in a centrifuge). By treating the extra acceleration terms due to the rotation of the frame as if they were forces, subtracting them from the physical forces, it's possible to treat the second time derivative of position (relative to the rotating frame) as if it was the absolute acceleration. Thus the analysis using Newton's law can proceed as if the reference frame was inertial, provided the fictitious force terms are included in the sum of forces. For example, centrifugal force is used in the FAA pilot's manual in describing turns. Other examples are such systems as planets, centrifuges, carousels, turning cars, spinning buckets, and rotating space stations. Regarding the advantages of rotating frames from the viewpoint of meteorology, Ryder says:

## Potential energy

In a reference frame uniformly rotating at angular rate Ω, the fictitious centrifugal force is conservative and has a potential energy of the form:

$E_p = -frac\left\{1\right\}\left\{2\right\} m Omega^2 r^2 ,$

where r is the radius from the axis of rotation. This result can be verified by taking the gradient of the potential to obtain the radially outward force:

$F_\left\{Cfgl\right\}$$= -frac\left\{partial \right\}\left\{partial r\right\} E_p$$= m Omega^2 r .$

The potential energy is useful, for example, in calculating the form of the water surface in a rotating bucket. Let the height of the water be $h\left(r\right),$: then the potential energy per unit mass contributed by gravity is $g h\left(r\right)$ (g = acceleration due to gravity) and the total potential energy per unit mass on the surface is $gh\left(r\right) - frac\left\{1\right\}\left\{2\right\}Omega^2 r^2,$. In a static situation (no motion of the fluid in the rotating frame), this energy is constant independent of position r. Requiring the energy to be constant, we obtain the parabolic form:

$h\left(r\right) = frac\left\{Omega^2\right\}\left\{2g\right\}r^2 + h\left(0\right) ,$

where $h\left(0\right)$ is the height at r = 0 (the axis). See Figure 1.

Similarly, the potential energy of the centrifugal force is a minor contributor to the complex 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.

## Examples

Below several examples illustrate both the inertial and rotating frames of reference, and the role of centrifugal force and its relation to Coriolis force in rotating frameworks. For more examples see Fictitious force.

### Using fictitious forces

It has been mentioned that to deal with motion in a rotating frame of reference, one alternative to a solution based upon translating everything into an inertial frame instead is to apply Newton's laws of motion in the rotating frame by adding pseudo-forces, and then working directly in the rotating frame. Next is a simple example of this method.

Figure 3 illustrates that a body that is stationary relative to the non-rotating inertial frame S' appears to be rotating when viewed from the rotating frame S, which is rotating at angular rate Ω. Therefore, application of Newton's laws to what looks like circular motion in the rotating frame S at a radius R, requires an inward centripetal force of −m Ω2 R to account for the apparent circular motion. According to observers in S, this centripetal force in the rotating frame is provided as a net force that is the sum of the radially outward centrifugal pseudo force m Ω2 R and the Coriolis force −2m Ω × vrot. To evaluate the Coriolis force, we need the velocity as seen in the rotating frame, vrot. According to the formulas in the Derivation section, this velocity is given by −Ω × R. Hence, the Coriolis force (in this example) is inward, in the opposite direction to the centrifugal force, and has the value −2m Ω2 R. The combination of the centrifugal and Coriolis force is then m Ω2 R−2m Ω2 R = −m Ω2 R, exactly the centripetal force required by Newton's laws for circular motion.

For further examples and discussion, see below, and see Taylor.

### Whirling table

Figure 4 shows a simplified version of an apparatus for studying centrifugal force called the "whirling table". The apparatus consists of a rod that can be whirled about an axis, causing a bead to slide on the rod under the influence of centrifugal force. A cord ties a weight to the sliding bead. By observing how the equilibrium balancing distance varies with the weight and the speed of rotation, the centrifugal force can be measured as a function of the rate of rotation and the distance of the bead from the center of rotation.

From the viewpoint of an inertial frame of reference, equilibrium results when the bead is positioned to select the particular circular orbit for which the weight provides the correct centripetal force.

The whirling table is a lab experiment, and standing there watching the table you have a detached viewpoint. It seems pretty much arbitrary whether to deal with centripetal force or centrifugal force. But if you were the bead, not the lab observer, and if you wanted to stay at a particular position on the rod, the centrifugal force would be how you looked at things. Centrifugal force would be pushing you around. Maybe the centripetal interpretation would come to you later, but not while you were coping with matters. Centrifugal force is not just mathematics.

### Rotating identical spheres

Figure 5 shows two identical spheres rotating about the center of the string joining them. This sphere example is one used by Newton himself to discuss the detection of rotation relative to absolute space. (A more practical experiment is to observe the isotropy of the cosmic background radiation.) The axis of rotation is shown as a vector Ω with direction given by the right-hand rule and magnitude equal to the rate of rotation: |Ω| = ω. The angular rate of rotation ω is assumed independent of time (uniform circular motion). Because of the rotation, the string is under tension. (See reactive centrifugal force.) The description of this system next is presented from the viewpoint of an inertial frame and from a rotating frame of reference.

#### Inertial frame

Adopt an inertial frame centered at the midpoint of the string. The balls move in a circle about the origin of our coordinate system. Look first at one of the two balls. To travel in a circular path, which is not uniform motion with constant velocity, but circular motion at constant speed, requires a force to act on the ball so as to continuously change the direction of its velocity. This force is directed inward, along the direction of the string, and is called a centripetal force. The other ball has the same requirement, but being on the opposite end of the string, requires a centripetal force of the same size, but opposite in direction. See Figure 6. These two forces are provided by the string, putting the string under tension, also shown in Figure 6.

#### Rotating frame

Adopt a rotating frame at the midpoint of the string. Suppose the frame rotates at the same angular rate as the balls, so the balls appear stationary in this rotating frame. Because the balls are not moving, observers say they are at rest. If they now apply Newton's law of inertia, they would say no force acts on the balls, so the string should be relaxed. However, they clearly see the string is under tension. (For example, they could split the string and put a spring in its center, which would stretch.) To account for this tension, they propose that in their frame a centrifugal force acts on the two balls, pulling them apart. This force originates from nowhere – it is just a "fact of life" in this rotating world, and acts on everything they observe, not just these spheres. In resisting this ubiquitous centrifugal force, the string is placed under tension, accounting for their observation, despite the fact that the spheres are at rest.

#### Coriolis force

What if the spheres are not rotating in the inertial frame (string tension is zero)? Then string tension in the rotating frame also is zero. But how can that be? The spheres in the rotating frame now appear to be rotating, and should require an inward force to do that. According to the analysis of uniform circular motion:

mathbf{F}_{mathrm{centripetal}} = -m mathbf{Omega times} left(mathbf{Omega times x_B }right)
$= -momega^2 R mathbf\left\{u\right\}_R ,$

where uR is a unit vector pointing from the axis of rotation to one of the spheres, and Ω is a vector representing the angular rotation, with magnitude ω and direction normal to the plane of rotation given by the right-hand rule, m is the mass of the ball, and R is the distance from the axis of rotation to the spheres (the magnitude of the displacement vector, |xB| = R, locating one or the other of the spheres). According to the rotating observer, shouldn't the tension in the string be twice as big as before (the tension from the centrifugal force plus the extra tension needed to provide the centripetal force of rotation)? The reason the rotating observer sees zero tension is because of yet another fictitious force in the rotating world, the Coriolis force, which depends on the velocity of a moving object. In this zero-tension case, according to the rotating observer the spheres now are moving, and the Coriolis force (which depends upon velocity) is activated. According to the article fictitious force, the Coriolis force is:


mathbf{F}_{mathrm{fict}} = - 2 m boldsymbolOmega times mathbf{v}_{B}

$= -2m omega left\left(omega R right\right) mathbf\left\{u\right\}_R ,$

where R is the distance to the object from the center of rotation, and vB is the velocity of the object subject to the Coriolis force, |vB| = ωR.

In the geometry of this example, this Coriolis force has twice the magnitude of the ubiquitous centrifugal force and is exactly opposite in direction. Therefore, it cancels out the ubiquitous centrifugal force found in the first example, and goes a step further to provide exactly the centripetal force demanded by uniform circular motion, so the rotating observer calculates there is no need for tension in the string − the Coriolis force looks after everything.

#### General case

What happens if the spheres rotate at one angular rate, say ωI (I = inertial), and the frame rotates at a different rate ωR (R = rotational)? The inertial observers see circular motion and the tension in the string exerts a centripetal inward force on the spheres of:

$mathbf\left\{T\right\} = -m omega_I^2 R mathbf\left\{u\right\}_R .$

This force also is the force due to tension seen by the rotating observers. The rotating observers see the spheres in circular motion with angular rate ωS = ωI − ωR (S = spheres). That is, if the frame rotates more slowly than the spheres, ωS > 0 and the spheres advance counterclockwise around a circle, while for a more rapidly moving frame, ωS < 0, and the spheres appear to retreat clockwise around a circle. In either case, the rotating observers see circular motion and require a net inward centripetal force:

$mathbf\left\{F\right\}_\left\{mathrm\left\{Centripetal\right\}\right\} = -m omega_S^2 R mathbf\left\{u\right\}_R .$

However, this force is not the tension in the string. So the rotational observers conclude that a force exists (which the inertial observers call a fictitious force) so that:

$mathbf\left\{F\right\}_\left\{mathrm\left\{Centripetal\right\}\right\} = mathbf\left\{T\right\} + mathbf\left\{F\right\}_\left\{mathrm\left\{Fict\right\}\right\} ,$

or,

$mathbf\left\{F\right\}_\left\{mathrm\left\{Fict\right\}\right\} = -m left\left(omega_S^2 R -omega_I^2 R right\right) mathbf\left\{u\right\}_R .$

The fictitious force changes sign depending upon which of ωI and ωS is greater. The reason for the sign change is that when ωI > ωS, the spheres actually are moving faster than the rotating observers measure, so they measure a tension in the string that actually is larger than they expect; hence, the fictitious force must increase the tension (point outward). When ωI < ωS, things are reversed so the fictitious force has to decrease the tension, and therefore has the opposite sign (points inward). Incidentally, checking the fictitious force needed to account for the tension in the string is one way for an observer to decide whether or not they are rotating – if the fictitious force is zero, they are not rotating. (Of course, in an extreme case like the gravitron amusement ride, you do not need much convincing that you are rotating, but standing on the Earth's surface, the matter is more subtle.)

##### Is the fictitious force ad hoc?
The introduction of FFict allows the rotational observers and the inertial observers to agree on the tension in the string. However, we might ask: "Does this solution fit in with general experience with other situations, or is it simply a "cooked up" ad hoc solution?" That question is answered by seeing how this value for FFict squares with the general result (derived in Fictitious force):


mathbf{F}_{mathrm{Fict}} = - 2 m boldsymbolOmega times mathbf{v}_{B} - m boldsymbolOmega times (boldsymbolOmega times mathbf{x}_B )  $- m frac\left\{d boldsymbolOmega \right\}\left\{dt\right\} times mathbf\left\{x\right\}_B .$

The subscript B refers to quantities referred to the non-inertial coordinate system. Full notational details are in Fictitious force. For constant angular rate of rotation the last term is zero. To evaluate the other terms we need the position of one of the spheres:

$mathbf\left\{x\right\}_B = Rmathbf\left\{u\right\}_R ,$

and the velocity of this sphere as seen in the rotating frame:

$mathbf\left\{v\right\}_B = omega_SR mathbf\left\{u\right\}_\left\{theta\right\} ,$

where uθ is a unit vector perpendicular to uR pointing in the direction of motion.

The vector of rotation Ω = ωR uz (uz a unit vector in the z-direction), and Ω × uR = ωR (uz × uR) = ωR uθ ; Ω × uθ = −ωR uR. The centrifugal force is then:

$mathbf\left\{F\right\}_mathrm\left\{Cfgl\right\} = - m boldsymbolOmega times \left(boldsymbolOmega times mathbf\left\{x\right\}_B \right) =momega_R^2 R mathbf\left\{u\right\}_R ,$
which naturally depends only on the rate of rotation of the frame and is always outward. The Coriolis force is
$mathbf\left\{F\right\}_mathrm\left\{Cor\right\} = - 2 m boldsymbolOmega times mathbf\left\{v\right\}_\left\{B\right\} = 2momega_S omega_R R mathbf\left\{u\right\}_R$

and has the ability to change sign, being outward when the spheres move faster than the frame (ωS > 0 ) and being inward when the spheres move slower than the frame (ωS < 0 ). Combining the terms:


mathbf{F}_{mathrm{Fict}} = mathbf{F}_mathrm{Cfgl} + mathbf{F}_mathrm{Cor} $=left\left(momega_R^2 R + 2momega_S omega_R Rright\right) mathbf\left\{u\right\}_R = momega_R left\left(omega_R + 2omega_S right\right) R mathbf\left\{u\right\}_R$
$=m\left(omega_I-omega_S\right)\left(omega_I+omega_S\right) R mathbf\left\{u\right\}_R = -m left\left(omega_S^2-omega_I^2right\right) R mathbf\left\{u\right\}_R .$
Consequently, the fictitious force found above for this problem of rotating spheres is consistent with the general result and is not an ad hoc solution just "cooked up" to bring about agreement for this single example. Moreover, it is the Coriolis force that makes it possible for the fictitious force to change sign depending upon which of ωI, ωS is the greater, inasmuch as the centrifugal force contribution always is outward.

### Dropping ball

Figure 7 shows a ball dropping vertically (parallel to the axis of rotation Ω of the rotating frame). For simplicity, suppose it moves downward at a fixed speed in the inertial frame, occupying successively the vertically aligned positions numbered one, two, three. In the rotating frame it appears to spiral downward, and the right side of Figure 7 shows a top view of the circular trajectory of the ball in the rotating frame. Because it drops vertically at a constant speed, from this top view in the rotating frame the ball appears to move at a constant speed around its circular track. A description of the motion in the two frames is next.

#### Inertial frame

In the inertial frame the ball drops vertically at constant speed. It does not change direction, so the inertial observer says the acceleration is zero and there is no force acting upon the ball.

#### Uniformly rotating frame

In the rotating frame the ball drops vertically at a constant speed, so there is no vertical component of force upon the ball. However, in the horizontal plane perpendicular to the axis of rotation, the ball executes uniform circular motion as seen in the right panel of Figure 7. Applying Newton's law of motion, the rotating observer concludes that the ball must be subject to an inward force in order to follow a circular path. Therefore, the rotating observer believes the ball is subject to a force pointing radially inward toward the axis of rotation. According to the analysis of uniform circular motion

mathbf{F}_{mathrm{fict}} = -momega^2 R ,

where ω is the angular rate of rotation, m is the mass of the ball, and R is the radius of the spiral in the horizontal plane. Because there is no apparent source for such a force (hence the label "fictitious"), the rotating observer concludes it is just "a fact of life" in the rotating world that there exists an inward force with this behavior. Inasmuch as the rotating observer already knows there is a ubiquitous outward centrifugal force in the rotating world, how can there be an inward force? The answer is again the Coriolis force: the component of velocity tangential to the circular motion seen in the right panel of Figure 7 activates the Coriolis force, which cancels the centrifugal force and, just as in the zero-tension case of the spheres, goes a step further to provide the centripetal force demanded by the calculations of the rotating observer. Some details of evaluation of the Coriolis force are shown in Figure 8.

Because the Coriolis force and centrifugal forces combine to provide the centripetal force the rotating observer requires for the observed circular motion, the rotating observer does not need to apply any additional force to the object, in complete agreement with the inertial observer, who also says there is no force needed. One way to express the result: the fictitious forces look after the "fictitious" situation, so the ball needs no help to travel the perceived trajectory: all observers agree that nothing needs to be done to make the ball follow its path.

### Parachutist

To show a different frame of reference, let's revisit the dropping ball example in Figure 7 from the viewpoint of a parachutist falling at constant speed to Earth (the rotating platform). The parachutist aims to land upon the point on the rotating ground directly below the drop-off point. Figure 9 shows the vertical path of descent seen in the rotating frame. The parachutist drops at constant speed, occupying successively the vertically aligned positions one, two, three.

In the stationary frame, let us suppose the parachutist jumps from a helicopter hovering over the destination site on the rotating ground below, and therefore traveling at the same speed as the target below. The parachutist starts with the necessary speed tangential to his path (ωR) to track the destination site. If the parachutist is to land on target, the parachute must spiral downward on the path shown in Figure 9. The stationary observer sees a uniform circular motion of the parachutist when the motion is projected downward, as in the left panel of Figure 9. That is, in the horizontal plane, the stationary observer sees a centripetal force at work, -m ω2 R, as is necessary to achieve the circular path. The parachutist needs a thruster to provide this force. Without thrust, the parachutist follows the dashed vertical path in the left panel of Figure 9, obeying Newton's law of inertia.

The stationary observer and the observer on the rotating ground agree that there is no vertical force involved: the parachutist travels vertically at constant speed. However, the observer on the ground sees the parachutist simply drop vertically from the helicopter to the ground, following the vertically aligned positions one, two, three. There is no force necessary. So how come the parachutist needs a thruster?

The ground observer has this view: there is always a centrifugal force in the rotating world. Without a thruster, the parachutist would be carried away by this centrifugal force and land far off the mark. From the parachutist's viewpoint, trying to keep the target directly below, the same appears true: a steady thrust radially inward is necessary, just to hold a position directly above target. Unlike the dropping ball case, where the fictitious forces conspired to produce no need for external agency, in this case they require intervention to achieve the trajectory. The basic rule is: if the inertial observer says a situation demands action or does not, the fictitious forces of the rotational frame will lead the rotational observer to the same conclusions, albeit by a different sequence.

Notice that there is no Coriolis force in this discussion, because the parachutist has zero horizontal velocity from the viewpoint of the ground observer.

## Development of the modern conception of centrifugal force

Early scientific ideas about centrifugal force were based upon intuitive perception, and circular motion was considered somehow more "natural" than straight line motion. According to Domenico Meli:
"For Huygens and Newton centrifugal force was the result of a curvilinear motion of a body; hence it was located in nature, in the object of investigation. According to a more recent formulation of classical mechanics, centrifugal force depends on the choice of how phenomena can be conveniently represented. Hence it is not located in nature, but is the result of a choice by the observer. In the first case a mathematical formulation mirrors centrifugal force; in the second it creates it."

There is evidence that Sir Isaac Newton originally conceived circular motion as being caused a balance between an inward centripetal force and an outward centrifugal force.

The modern conception of centrifugal force appears to have its origins in Christiaan Huygens' paper De Vi Centrifuga, written in 1659. It has been suggested that the idea of circular motion as caused by a single force was introduced to Newton by Robert Hooke.

Newton described the role of centrifugal force upon the height of the oceans near the equator in the :

The effect of centrifugal force in countering gravity, as in this behavior of the tides, has led centrifugal force sometimes to be called "false gravity" or "imitation gravity" or "quasi-gravity".

### Role in developing the idea of inertial frames

A continuing theme in classical mechanics has been the role of "absolute space". In the rotating bucket experiment Newton observed the shape of the surface of water in a bucket as the bucket was spun on a rope. At first the water is flat, then, as it acquires the same rotation as the bucket, it becomes parabolic. This shape is a consequence of centrifugal force, see subsection Potential energy above. Newton took this change as evidence that one could detect motion relative to "absolute space" experimentally, in this instance by looking at the shape of the surface of the water.

Later scientists found this view unwarranted: they pointed out (as did Newton) that the laws of mechanics were the same for all observers that differed only by uniform translation; that is, all observers that differed in motion only by a constant velocity. Hence, the "fixed stars" or "absolute space" was not preferred, but only one of a set of frames related by Galilean transformations. The inadequacy of the notion of "absolute space" in Newtonian mechanics is spelled out by Blagojević:

Ultimately this notion of the transformation properties of physical laws between frames played a more and more central role. It was noted that accelerating frames exhibited "fictitious forces" like the centrifugal force. These forces did not behave under transformation like other forces, providing a means of distinguishing them. This peculiarity of these forces led to the names inertial forces, pseudo-forces or fictitious forces. In particular, fictitious forces did not appear at all in some frames: those frames differing from that of the fixed stars by only a constant velocity. Thus, the preferred frames, called "inertial frames", were identifiable by the absence of fictitious forces.

The idea of an inertial frame was extended further in the special theory of relativity. This theory posited that all physical laws should appear of the same form in inertial frames, not just the laws of mechanics. In particular, Maxwell's equations should apply in all frames. Because Maxwell's equations implied the same speed of light in the vacuum of free space for all inertial frames, inertial frames now were found to be related not by Galilean transformations, but by Poincaré transformations, of which a subset is the Lorentz transformations. That posit led to many ramifications, including Lorentz contractions and relativity of simultaneity. Einstein succeeded, through many clever thought experiments, in showing that these apparently odd ramifications in fact had very natural explanation upon looking at just how measurements and clocks actually were used. That is, these ideas flowed from operational definitions of measurement coupled with the experimental confirmation of the constancy of the speed of light.

Later the general theory of relativity further generalized the idea of frame independence of the laws of physics, and abolished the special position of inertial frames, at the cost of introducing curved space-time. Following an analogy with centrifugal force (sometimes called "artificial gravity" or "false gravity"), gravity itself became a fictitious force, as enunciated in the principle of equivalence.

In short, centrifugal force played a key early role in establishing the set of inertial frames of reference and the significance of fictitious forces, even aiding in the development of general relativity.

## 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. Inertial drum brake ascenders used in rock climbing and the inertia reels used in many automobile seat belts operate on the same principle.
• 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.
• 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.

• 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.
• When spinning sufficiently fast, centrifugal force can destroy a CD

Nevertheless, all of these systems can also be described without requiring the concept of centrifugal force, 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.