Centrifugal force

 

Zobacz też:
The factual accuracy of this article is disputed.
Please see the relevant discussion on the talk page. (August 2008)

For centrifugal force that isn't due to rotating reference frames, see centrifugal force (disambiguation).
For the external force required to make a body follow a curved path, see Centripetal force.
For general derivations and discussion of fictitious forces, see Fictitious force.

In classical mechanics, centrifugal force (from Latin centrum "center" and fugere "to flee") is one of the three so-called inertial forces or fictitious forces that enter the equations of motion when Newton's laws are formulated in a rotating reference frame. The other two fictitious forces are the Coriolis force and the Euler force.[1] Any object, viewed from a rotating frame, is subject to a centrifugal force which depends only on the position and the mass of the object, and is oriented outward from the axis of rotation of the rotating frame.[2][3][4]

Unlike real forces such as electromagnetic forces, fictitious forces do not originate from physical interactions between objects.

The apparent motion that may be ascribed to centrifugal force is sometimes called the centrifugal effect. [5][6]

Contents

Analysis using fictitious forces

Fictitious forces do not appear in the equations of motion in an inertial frame of reference: in an inertial frame, the motion of an object is explained by the real impressed forces. In a non-inertial frame such as a rotating frame, however, Newton's first and second laws still can be used to make accurate physical predictions provided fictitious forces are included along with the real forces. For solving problems of mechanics in non-inertial reference frames, the advice given in textbooks is:[7] [8]

Treat the fictitious forces like real forces, and pretend you are in an inertial frame.

– Louis N. Hand, Janet D. Finch Analytical Mechanics, p. 267

Because fictitious forces do not originate from other objects, there is no originating object to experience an associated reaction force: Newton's third law does not apply to fictitious forces.

Advantages of rotating frames

A rotating reference frame can have advantages over an inertial reference frame.[3][4] 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). With the addition of centrifugal force and other fictitious forces, Newton's first and second laws can be used to think about these systems, and to calculate motions within them. A specific example: centrifugal force is used in the FAA pilot's manual in describing turns.[9] Other examples are such systems as planets, centrifuges, carousels, turning cars, spinning buckets, and rotating space stations.[10][11][12]

Choice of observational frame of reference

Centrifugal force can be confusing.[13] The ideas of Newtonian mechanics had to overcome numerous "common sense" perceptions, among them the belief that friction is an inherent property of motion rather than an externally applied force,[14][15] and the failure to recognize that change in the direction of motion can be as important as change in speed (that is, the concept of velocity as a vector quantity).[16] In particular, an object in circular motion must continuously change direction. This directional change requires an inward centripetal force that keeps the object on its circular path by changing the direction of its velocity; without the centripetal force, the object would follow a straight trajectory. However, intuition and common sense may disagree with this viewpoint. A source of confusion is the instinctive adoption of a reference frame, which can be unconscious, as in the example explored below.

Consider a car going around a turn. A passenger may experience centrifugal force: as the car turns, the passenger feels pushed against the door by a force acting toward the outside of the curve. That interpretation of experience is the view from one reference frame: a non-inertial, rotating reference frame. Experience is interpreted in terms of the fictitious centrifugal force. Contrast this view with the description of the same events in terms of an inertial coordinate system, according to which the passenger tends to travel in a straight line, but because the car is going in a circle, it pushes the passenger inward (not outward) to keep them turning. In accordance with Newton's third law, the passenger applies an outward reaction force to the car door, but no outward force acts on the passenger; the force the passenger feels pushing them outward against the door disappears in an inertial frame. Both descriptions are valid in the sense that either will lead to a successful design of the catch on the car door to retain the passenger inside the vehicle without popping open, and either description will lead to a correct banking of the curve on the road. Further discussion of this example can be found in the article on reactive centrifugal force.

As another illustration of the difference between reference frames, suppose we swing a ball around our head on a string. A natural viewpoint is that the ball is pulling on the string, and we have to resist that pull or the ball will fly away. That perspective puts us in a rotating frame of reference – we are reacting to the ball and have to fight centrifugal force. A less intuitive frame of mind is that we have to keep pulling on the ball, or else it will not change direction to stay in a circular path. That is, we are in an active frame of mind: we have to supply centripetal force. That puts us in an inertial frame of reference.

The centrifuge supplies another example, where often the rotating frame is preferred and centrifugal force is treated explicitly.[17] This example can become more complicated than the ball on string, however, because there may be forces due to friction, buoyancy, and diffusion; not just the fictitious forces of rotational frames. The balance between dragging forces like friction and driving forces like the centrifugal force is called sedimentation. A complete description leads to the Lamm equation.[18][19]

Intuition can go either way, and we can become perplexed when we switch viewpoints unconsciously. Standard physics teaching is often ineffective in clarifying these intuitive perceptions,[13][20] and beliefs about centrifugal force (and other such forces) grounded in the rotating frame often remain fervently held as somehow real regardless of framework, despite the classical explanation that such descriptions always are framework dependent.[13][20][21]

Are centrifugal and Coriolis forces "real"?

See also: Gravitron

The adjective "fictitious" when used to describe forces like the centrifugal force is used in a technical sense and is not meant to suggest that the forces are not "real" in the everyday sense of the adjective "real". Below the technical meaning is explained further and some examples are provided where the reality of fictitious forces is a common experience.

Newton viewed his law of inertia as valid in any reference frame moving with uniform velocity relative to the fixed stars;[22] that is, neither rotating nor accelerating relative to the stars.[23] Today the notion of "absolute space" is abandoned, and an inertial frame of reference in the field of classical non-relativistic mechanics is defined as:[24][25]

An inertial frame of reference is one in which the motion of a particle not subject to forces is in a straight line at constant speed.

Here, forces refers to forces originating in identifiable bodes, such as electrical charges. The distinction between real and fictitious forces is important in physics, where the study of interactions between bodies is a major topic, for example, in the standard model and the theory of everything.

The centrifugal and Coriolis forces are called fictitious because they do not appear in an inertial frame of reference, but instead are present only in non-inertial (most commonly, rotating) frames. In effect, one can identify an inertial frame by the observation that morphing fictitious forces are not present, forces that change form from one reference frame to another (and disappear altogether in an inertial frame). (An example is the rotating spheres example below, where the fictitious forces are shown to disappear in an inertial frame and to take on a variety of forms in other frames.) Besides their variation from frame to frame, fictitious forces can be identified because, unlike real forces, they do not originate in objects in the environment.[26] Apart from fundamental issues of physics, for purposes of the study of motion of bodies, fictitious forces are treated as real by observers within accelerating or rotating environments.

An interesting exploration of the apparent reality of centrifugal forces is provided by the artificial gravity introduced into a space station by rotation.[27] Such a form of gravity does have things in common with ordinary gravity. For example, playing catch, the ball must be thrown upward to counteract "gravity". Cream will rise to the top of milk (if it is not homogenized). There are differences from ordinary gravity: one is the rapid change in "gravity" with distance from the center of rotation, which would be very noticeable unless the space station were very large. More disconcerting is the associated Coriolis force, which introduces signals in the ear that conflict with vision, leading to nausea.[28][29] Differences between artificial and real gravity can affect human health, and are a subject of study.[30] In any event, the fictitious forces in this habitat would seem perfectly real to those living in the station. Although they readily could do experiments to reveal the space station was rotating, the inhabitants of the space station would find description of daily life remained more natural in terms of fictitious forces.

From a time-honored viewpoint,[31] the simplest explanation is often to be preferred. The simplest explanation often involves fictitious forces.

Fictitious forces

Main article: Fictitious force
Figure 1: Object stationary in inertial frame S' as observed in rotating frame S. Top panel: In inertial (stationary) frame S' , frame S is rotating counterclockwise at angular rate ω, and occupies successive counterclockwise positions at times t0, t1, and t2. Stationary object does not move, of course. Center panel: Positions of the stationary object as it appears in S at the times t0, t1, and t2. The object appears to move clockwise in S. Bottom panel: Assembly of the positions in center panel to construct the orbit of the stationary object as seen by S. Radius vectors from the origin of moving frame S to the object at times t0, t1, t2 are R0, R1, R2; these vectors all have magnitude equal to the radius of the circle R. At time t0, the object has a velocity v0 in frame S, but this velocity turns with motion of the object to remain tangential to its orbit at all times.
Figure 1: Object stationary in inertial frame S' as observed in rotating frame S. Top panel: In inertial (stationary) frame S' , frame S is rotating counterclockwise at angular rate ω, and occupies successive counterclockwise positions at times t0, t1, and t2. Stationary object does not move, of course. Center panel: Positions of the stationary object as it appears in S at the times t0, t1, and t2. The object appears to move clockwise in S. Bottom panel: Assembly of the positions in center panel to construct the orbit of the stationary object as seen by S. Radius vectors from the origin of moving frame S to the object at times t0, t1, t2 are R0, R1, R2; these vectors all have magnitude equal to the radius of the circle R. At time t0, the object has a velocity v0 in frame S, but this velocity turns with motion of the object to remain tangential to its orbit at all times.

An alternative to dealing with a rotating frame of reference from the inertial standpoint is to make Newton's laws of motion valid in the rotating frame by artificially adding pseudo forces to be the cause of the above acceleration terms, and then working directly in the rotating frame.[2][32][33]

Here attention is restricted to frames rotating about a fixed axis. (For a discussion of complex rotations of a rigid body, see Euler angles.) In such frames, the centrifugal acceleration is added to the motion of every object, and attributed to a centrifugal force, given by:

\mathbf{F}_\mathrm{centrifugal} \, = m \mathbf{a}_\mathrm{centrifugal} \,
=m \omega^2 \mathbf{R} \,

where m\, is the mass of the object, ω = dθ / dt is the angular rate of rotation, and R is the vector that locates the object relative to the center of rotation (R is perpendicular to the axis of rotation and points outward to the location of the rotating 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":[34]

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

where vrot is the velocity as seen in the rotating frame of reference and × denotes the vector cross product. The rotation vector Ω points along the axis of rotation of the rotating frame with direction given by the right-hand rule and has magnitude ω, the angular rate of rotation.

Figure 1 is an example. A body that is stationary relative to the non-rotating inertial frame S' will be rotating when viewed from the rotating frame S. Therefore, Newton's laws, as applied in S to what looks like circular motion in the rotating frame, requires an inward centripetal force of −m ω2 R to account for the apparent circular motion. This centripetal force in the rotating frame is provided as the sum of the radially outward centrifugal pseudo force m ω2 R and the Coriolis force −2m Ω × vrot.[35] [36] To evaluate the Coriolis force, we need the velocity as seen in the rotating frame. Some pondering[37] will show that 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.[38] [39][40]

For further examples and discussion, see below, and see Taylor.[41]

Because this centripetal force is combined from only pseudo forces, it is "fictitious" in the sense of having no apparent origin from physical sources (unlike electrical, magnetic or gravitational fields, which are produced by bodies in the environment), the combination of pseudo forces simply is posited as a "fact of life" in the rotating frame, it is just "there". It has to be included as a force in Newton's laws if calculations of trajectories in the rotating frame are to come out right.

Moving objects and observational frames of reference

Figure 2: Local coordinate system for planar motion on a curve. Two different positions are shown for distances s and s + ds along the curve. At each position s, unit vector un points along the outward normal to the curve and unit vector ut is tangential to the path. The radius of curvature of the path is ρ as found from the rate of rotation of the tangent to the curve with respect to arc length, and is the radius of the osculating circle at position s. The unit circle on the left shows the rotation of the unit vectors with s.
Figure 2: Local coordinate system for planar motion on a curve. Two different positions are shown for distances s and s + ds along the curve. At each position s, unit vector un points along the outward normal to the curve and unit vector ut is tangential to the path. The radius of curvature of the path is ρ as found from the rate of rotation of the tangent to the curve with respect to arc length, and is the radius of the osculating circle at position s. The unit circle on the left shows the rotation of the unit vectors with s.
See also: Generalized forces, Generalized force, Curvilinear coordinates, Generalized coordinates, and Frenet-Serret formulas

In discussion of a particle moving in a circular orbit,[42] one can identify the centripetal and tangential forces. It then seems to be no problem to switch hats and talk about the fictitious centrifugal and Euler forces. But what underlies this switch is a change of observational frame of reference from the inertial frame where we started, where centripetal and tangential forces make sense, to a rotating frame of reference where the particle appears motionless and fictitious centrifugal and Euler forces have to be brought into play. That switch is unconscious, but real.

Suppose we sit on a particle in planar motion. What switch of hats leads to fictitious centrifugal and Euler forces?

To explore that question, begin in a stationary frame of reference. Consider using a coordinate system commonly used in planar motion, the so-called local coordinate system.[43] See Figure 2.

To introduce the unit vectors of the local coordinate system shown in Figure 2, one approach is to begin in Cartesian coordinates in an inertial framework and describe the local coordinates in terms of these Cartesian coordinates. In Figure 2, the arc length s is the distance the particle has traveled along its path in time t. The path r (t) with components x(t), y(t) in Cartesian coordinates is described using arc length s(t) as:[44]

\mathbf{r}(s) = \left[ x(s),\ y(s) \right] \ .

One way to look at the use of s is to think of the path of the particle as sitting in space, like the trail left by a skywriter, independent of time. Any position on this path is described by stating its distance s from some starting point on the path. Then an incremental displacement along the path ds is described by:

d\mathbf{r}(s) = \left[ dx(s),\ dy(s) \right]=\left[ x'(s),\ y'(s) \right] ds \ ,

where primes are introduced to denote derivatives with respect to s. The magnitude of this displacement is ds, showing that:[45]

\left[ x'(s)^2 + y'(s)^2 \right] = 1 \ .     (Eq. 1)

This displacement is necessarily tangent to the curve at s, showing that the unit vector tangent to the curve is:

\mathbf{u}_t(s) = \left[ x'(s), \ y'(s) \right] \ ,

while the outward unit vector normal to the curve is

\mathbf{u}_n(s) = \left[ y'(s),\ -x'(s) \right] \ ,

Orthogonality can be verified by showing the vector dot product is zero. The unit magnitude of these vectors is a consequence of Eq. 1.

As an aside, notice that the use of unit vectors that are not aligned along the Cartesian xy-axes does not mean we are no longer in an inertial frame. All it means is that we are using unit vectors that vary with s to describe the path, but still observe the motion from the inertial frame.

Using the tangent vector, the angle of the tangent to the curve, say θ, is given by:

\sin \theta =\frac{y'(s)}{\sqrt{x'(s)^2+y'(s)^2}} = y'(s) \ ;   and   \cos \theta =\frac{x'(s)}{\sqrt{x'(s)^2+y'(s)^2}} = x'(s) \ .

The radius of curvature is introduced completely formally (without need for geometric interpretation) as:

\frac{1}{\rho} = \frac{d\theta}{ds}\ .

The derivative of θ can be found from that for sin θ:

\frac{d \sin\theta}{ds} = \cos \theta \frac {d\theta}{ds} = \frac{1}{\rho} \cos \theta \
= \frac{1}{\rho} x'(s)\ .

Now:

\frac{d \sin \theta }{ds} = \frac{d}{ds} \frac{y'(s)}{\sqrt{x'(s)^2+y'(s)^2}}  = \frac{y''(s)x'(s)^2-y'(s)x'(s)x''(s)} {\left(x'(s)^2+y'(s)^2\right)^{3/2}}\ ,

in which the denominator is unity according to Eq. 1. With this formula for the derivative of the sine, the radius of curvature becomes:

\frac {d\theta}{ds} = \frac{1}{\rho} = y''(s)x'(s) - y'(s)x''(s)\  =\frac{y''(s)}{x'(s)} = -\frac{x''(s)}{y'(s)} \ ,

where the equivalence of the forms stems from differentiation of Eq. 1:

x'(s)x''(s) + y'(s)y''(s) = 0 \ .

Having set up the description of any position on the path in terms of its associated value for s, and having found the properties of the path in terms of this description, motion of the particle is introduced by stating the particle position at any time t as the corresponding value s (t).

Using the above results for the path properties in terms of s, the acceleration in the inertial reference frame as described in terms of the components normal and tangential to the path of the particle can be found in terms of the function s(t) and its various time derivatives (as before, primes indicate differentiation with respect to s):

\mathbf{a}(s) = \frac{d}{dt}\mathbf{v}(s)   = \frac{d}{dt}\left[\frac{ds}{dt} \left( x'(s), \ y'(s) \right) \right]\
= \left(\frac{d^2s}{dt^2}\right)\mathbf{u}_t(s) +\left(\frac{ds}{dt}\right) ^2 \left(x''(s),\ y''(s) \right)
= \left(\frac{d^2s}{dt^2}\right)\mathbf{u}_t(s) - \left(\frac{ds}{dt}\right) ^2 \frac{1}{\rho} \mathbf{u}_n(s) \ ,

as can be verified by taking the dot product with the unit vectors ut(s) and un(s). This result for acceleration is the same as that for circular motion based on the radius ρ. Using this coordinate system in the inertial frame, it is easy to identify the force normal to the trajectory as the centripetal force and that parallel to the trajectory as the tangential force.

Next, we change observational frames. Sitting on the particle, we adopt a non-inertial frame where the particle is at rest (zero velocity). This frame has a continuously changing origin, which at time t is the center of curvature of the path at time t, and whose rate of rotation is the angular rate of motion of the particle about that origin at time t. This non-inertial frame also employs unit vectors normal to the trajectory and parallel to it. The angular velocity of this frame is the angular velocity of the particle about the center of curvature at time t. The centripetal force of the inertial frame becomes the force necessary to overcome the centrifugal force in the non-inertial frame where the body is at rest. Likewise, the force causing any acceleration of speed along the path seen in the inertial frame becomes the force necessary to overcome the Euler force in the non-inertial frame where the particle is at rest. There is zero Coriolis force in the frame, because the particle has zero velocity in this frame. For a pilot in an airplane, for example, these fictitious forces are a matter of direct experience.[46] However, these fictitious forces cannot be related to a simple observational frame of reference other than the particle itself, unless it is in a particularly simple path, like a circle.

That said, from a qualitative standpoint, the path of an airplane can be approximated by an arc of a circle for a limited time, and for the limited time a particular radius of curvature applies, the centrifugal and Euler forces can be analyzed on the basis of circular motion with that radius. See article discussing turning an airplane.

Next, reference frames rotating about a fixed axis are discussed in more detail.

Uniformly rotating reference frames

See also: Circular motion, Uniform circular motion, and Inertial reference frame

Rotating reference frames are used in physics, mechanics, or meteorology whenever they are the most convenient frame to use.

The laws of physics are the same in all inertial frames. But a rotating reference frame is not an inertial frame, so the laws of physics are transformed from the inertial frame to the rotating frame. For example, assuming a constant rotation speed, transformation is achieved by adding to every object two coordinate accelerations that correct for the constant rotation of the coordinate axes. The vector equations describing these accelerations are (see fictitious force for a derivation):[26][47][4]

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

where \mathbf{a}_\mathrm{rot}\, is the acceleration relative to the rotating frame, \mathbf{a}\, is the acceleration relative to the inertial frame, \mathbf{\Omega}\, is the angular velocity vector describing the rotation of the reference frame,[48] \mathbf{v_\mathrm{rot}}\, is the velocity of the body relative to the rotating frame, and \mathbf{r}\, is the position vector of the body. The last term is the centrifugal acceleration:

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

where R is the component of \mathbf{r}\, perpendicular to the axis of rotation.

Non uniformly rotating reference frame

Main article: Accelerated reference frame

Although changing coordinates from an inertial frame of reference to any rotating one alters the equations of motion to require the inclusion of two sources of fictitious force, the centrifugal force, and the Coriolis force,[3][4] a third term, the Euler acceleration must be added if the rotation of the frame varies,[49] and a fourth acceleration is needed if the frame is linearly accelerating.[50]

Potential energy

Figure 3: The interface of two immiscible liquids rotating around a vertical axis is an upward-opening circular paraboloid.
Figure 3: The interface of two immiscible liquids rotating around a vertical axis is an upward-opening circular paraboloid.
Figure 4: Potential energy contours for the restricted three-body problem. Points of equilibrium (Lagrange points) are labeled on a contour plot of the effective potential of a two-body system (the Sun and Earth here, treating the Moon as a negligible mass) as viewed from a rotating frame of reference that is centered at the center of mass in which Sun and Earth remain stationary. The total potential is the sum of the potential energy due to the centrifugal force as well as the potential due to gravity. The contours show equipotential surfaces. The arrows indicate the gradients of increasing potential around the five Lagrange points — toward or away from them, but at the points themselves these forces are balanced. See NASA Wilkinson Microwave Anisotropy Probe for more details
Figure 4: Potential energy contours for the restricted three-body problem. Points of equilibrium (Lagrange points) are labeled on a contour plot of the effective potential of a two-body system (the Sun and Earth here, treating the Moon as a negligible mass) as viewed from a rotating frame of reference that is centered at the center of mass in which Sun and Earth remain stationary. The total potential is the sum of the potential energy due to the centrifugal force as well as the potential due to gravity. The contours show equipotential surfaces. The arrows indicate the gradients of increasing potential around the five Lagrange points — toward or away from them, but at the points themselves these forces are balanced. See NASA Wilkinson Microwave Anisotropy Probe for more details
See also: Bucket argument and Coriolis effect

In a uniformly rotating reference frame, the fictitious centrifugal force is conservative and has a potential energy of the form

E_p = -\frac{1}{2} 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:

FCfgl = -\frac{\partial }{\partial r} 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(r)\,: then the potential energy per unit mass contributed by gravity is g h(r) \ (g = acceleration due to gravity) and the total potential energy per unit mass on the surface is gh(r) - \frac{1}{2}\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(r) = \frac{\omega^2}{2g}r^2 + h(0) \ ,

where h(0) is the height at r = 0 (the axis). See Figure 3.

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.

Whirling table

Figure 5: The "whirling table". The rod is made to rotate about the axis and the centrifugal force of the sliding bead is balanced by the weight attached by a cord over two pulleys.
Figure 5: The "whirling table". The rod is made to rotate about the axis and the centrifugal force of the sliding bead is balanced by the weight attached by a cord over two pulleys.

Figure 5 shows a simplified version of an apparatus for studying centrifugal force called the "whirling table".[51] 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 6: Two spheres tied with a string and rotating at an angular rate ω. Because of the rotation, the string tying the spheres together is under tension.
Figure 6: Two spheres tied with a string and rotating at an angular rate ω. Because of the rotation, the string tying the spheres together is under tension.
Figure 7: Exploded view of rotating spheres in an inertial frame of reference showing the centripetal forces on the spheres provided by the tension in the tying string.
Figure 7: Exploded view of rotating spheres in an inertial frame of reference showing the centripetal forces on the spheres provided by the tension in the tying string.

Figure 6 shows two identical spheres rotating about the center of the string joining them. This sphere example is one used by Newton himself.[52] 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 7. These two forces are provided by the string, putting the string under tension, also shown in Figure 7.

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

Main article: Coriolis effect

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:[35]

\mathbf{F}_{\mathrm{centripetal}} = -m \mathbf{\Omega \ \times} \left( \mathbf{\Omega \times x_B }\right) \
= -m\omega^2 R\ \mathbf{u}_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:[35]

\mathbf{F}_{\mathrm{fict}} = - 2 m \boldsymbol\Omega \times \mathbf{v}_{B} \
= -2m \omega \left( \omega R \right)\ \mathbf{u}_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{T} = -m \omega_I^2 R \mathbf{u}_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{F}_{\mathrm{Centripetal}} = -m \omega_S^2 R \mathbf{u}_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{F}_{\mathrm{Centripetal}} = \mathbf{T} + \mathbf{F}_{\mathrm{Fict}}\ ,

or,

\mathbf{F}_{\mathrm{Fict}} = -m \left( \omega_S^2 R -\omega_I^2 R \right) \mathbf{u}_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):[53]

\mathbf{F}_{\mathrm{Fict}} = - 2 m \boldsymbol\Omega \times \mathbf{v}_{B} - m \boldsymbol\Omega \times (\boldsymbol\Omega \times \mathbf{x}_B )  \ - m \frac{d \boldsymbol\Omega }{dt} \times \mathbf{x}_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{x}_B = R\mathbf{u}_R \ ,

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

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

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{F}_\mathrm{Cfgl} = - m \boldsymbol\Omega \times (\boldsymbol\Omega \times \mathbf{x}_B ) =m\omega_R^2 R \mathbf{u}_R\ ,

which naturally depends only on the rate of rotation of the frame and is always outward. The Coriolis force is

\mathbf{F}_\mathrm{Cor} = - 2 m \boldsymbol\Omega \times \mathbf{v}_{B} = 2m\omega_S \omega_R R \mathbf{u}_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 ).[54] Combining the terms:

\mathbf{F}_{\mathrm{Fict}} = \mathbf{F}_\mathrm{Cfgl} + \mathbf{F}_\mathrm{Cor}  =\left( m\omega_R^2 R + 2m\omega_S \omega_R R\right) \mathbf{u}_R = m\omega_R \left( \omega_R + 2\omega_S \right) R \mathbf{u}_R
=m(\omega_I-\omega_S)(\omega_I+\omega_S)\ R \mathbf{u}_R = -m \left(\omega_S^2-\omega_I^2\right)\ R \mathbf{u}_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 8: A ball moving vertically along the axis of rotation in an inertial frame appears to spiral downward in the rotating frame. The right panel shows a downward view in the rotating frame. The rate of rotation |Ω| = ω is assumed constant in time.
Figure 8: A ball moving vertically along the axis of rotation in an inertial frame appears to spiral downward in the rotating frame. The right panel shows a downward view in the rotating frame. The rate of rotation |Ω| = ω is assumed constant in time.
Figure 9: Vector cross product used to determine the Coriolis force. The vector Ω represents the rotation of the frame at angular rate ω; the vector v shows the velocity tangential to the circular motion as seen in the rotating frame. The vector Ω × v is found using the right-hand rule for vector cross products. It is related to the negative of the Coriolis force (the Coriolis force is −2 m Ω × v).
Figure 9: Vector cross product used to determine the Coriolis force. The vector Ω represents the rotation of the frame at angular rate ω; the vector v shows the velocity tangential to the circular motion as seen in the rotating frame. The vector Ω × v is found using the right-hand rule for vector cross products. It is related to the negative of the Coriolis force (the Coriolis force is −2 m Ω × v).

Figure 8shows 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 8 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 8. 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}} = -m\omega^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 8 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 9.

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

Figure 10: A parachutist moving vertically parallel to the axis of rotation in a rotating frame appears to spiral downward in the inertial frame. The parachutist begins the drop with a horizontal component of velocity the same as the target site. The left panel shows a downward view in the inertial frame. The rate of rotation |Ω| = ω is assumed constant in time.
Figure 10: A parachutist moving vertically parallel to the axis of rotation in a rotating frame appears to spiral downward in the inertial frame. The parachutist begins the drop with a horizontal component of velocity the same as the target site. The left panel shows a downward view in the inertial frame. The rate of rotation |Ω| = ω is assumed constant in time.

To show a different frame of reference, let's revisit the dropping ball example in Figure 8 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 10 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 10. The stationary observer sees a uniform circular motion of the parachutist when the motion is projected downward, as in the left panel of Figure 10. 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 10, 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.

Centrifugal force in polar coordinates

Main article: polar coordinates

First, a brief orientation is provided to point out that time varying coordinates are used in both inertial and non-inertial frames of reference, so the use of time varying coordinates should not be confounded with a change of observer, but is only a change of the observer's choice of description. Then polar coordinates are introduced for use in (first) an inertial frame of reference and then (second) in a rotating frame of reference.

Time varying coordinate systems

In a general coordinate system, the basis vectors for the coordinates may vary in time at fixed positions, or they may vary with position at fixed times, or both. It may be noted that coordinate systems attached to both inertial frames and non-inertial frames can have basis vectors that vary in time, space or both, for example the description of a trajectory in polar coordinates as seen from an inertial frame.[55] or as seen from a rotating frame.[56]

Here are two quotes relating "state of motion" and "coordinate system":[57][58]

We first introduce the notion of reference frame, itself related to the idea of observer: the reference frame is, in some sense, the "Euclidean space carried by the observer". Let us give a more mathematical definition:… the reference frame is... the set of all points in the Euclidean space with the rigid body motion of the observer. The frame, denoted \mathfrak{R}, is said to move with the observer.… The spatial positions of particles are labelled relative to a frame \mathfrak{R} by establishing a coordinate system R with origin O. The corresponding set of axes, sharing the rigid body motion of the frame \mathfrak{R}, can be considered to give a physical realization of \mathfrak{R}. In a frame \mathfrak{R}, coordinates are changed from R to R' by carrying out, at each instant of time, the same coordinate transformation on the components of intrinsic objects (vectors and tensors) introduced to represent physical quantities in this frame.

– Jean Salençon, Stephen Lyle. (2001). Handbook of Continuum Mechanics: General Concepts, Thermoelasticity p. 9

In traditional developments of special and general relativity it has been customary not to distinguish between two quite distinct ideas. The first is the notion of a coordinate system, understood simply as the smooth, invertible assignment of four numbers to events in spacetime neighborhoods. The second, the frame of reference, refers to an idealized system used to assign such numbers … To avoid unnecessary restrictions, we can divorce this arrangement from metrical notions. … Of special importance for our purposes is that each frame of reference has a definite state of motion at each event of spacetime.

– John D. Norton: General Covariance and the Foundations of General Relativity: eight decades of dispute, Rep. Prog. Phys., 56, pp. 835-6.

Two terminologies

When equations of motion are expressed in terms of any curvilinear coordinate system, extra terms appear. Strictly speaking, these terms represent how the basis vectors change as the coordinates change. In a purely mathematical treatment, regardless of the frame that the coordinate system is associated with (inertial or non-inertial), the extra terms lead to terms in the acceleration of an observed particle that sometimes are referred to as fictitious forces. According to this terminology, fictitious forces are determined by the coordinate system itself, regardless of the frame it is attached to, that is regardless of whether the coordinate system is attached to an inertial or a non-inertial frame of reference. In contrast, the fictitious forces related to state of motion of the observer vanish in inertial frames of reference. To distinguish these two terminologies, the fictitious forces that vanish in an inertial frame of reference are called in this article the "state-of-motion" fictitious forces and those that originate in the interpretation of the form of time derivatives in particular coordinate systems are called "coordinate" fictitious forces.

Assuming it is clear that "state of motion" and "coordinate system" are different, it follows that the dependence of centrifugal force (as in this article) upon "state of motion" and its independence from "coordinate system", which contrasts with the "coordinate" version with exactly the opposite dependencies, indicates that two different ideas are referred to by the terminology "fictitious force". The present article is about one of these two ideas ("state-of-motion"), not both of them.

Polar coordinates in an inertial frame of reference

Below, the "state-of-motion" fictitious forces defined for non-inertial frames are related to the "coordinate" versions for polar coordinates. In an inertial frame, let \mathbf{r} be the position vector of a moving particle. Its Cartesian components (x, y) are:

\mathbf{r} =(r\cos\theta,\ r\sin\theta)\ ,

with r and θ depending on time t.

Unit vectors are defined in the radially outward direction \mathbf{r}:

\hat{\mathbf{r}}=(\cos\theta,\ \sin\theta)

and in the direction at right angles to \mathbf{r}:

\hat{\boldsymbol\theta}=(-\sin\theta\ ,\cos\theta) \ .

These unit vectors vary in direction with time:

\frac{d}{dt}\hat{\boldsymbol{r}} = (-\sin\theta,\ \cos\theta)\frac{d \theta}{dt} = \frac{d \theta}{dt}\hat{\boldsymbol\theta} ,

and:

\frac{d}{dt}\hat{\boldsymbol{\theta}} = (-\cos\theta,\ -\sin\theta)\frac{d \theta}{dt} =- \frac{d \theta}{dt}\hat{\boldsymbol r} .

Using these derivatives, the first and second derivatives of position are:

\boldsymbol{v} =\frac{d\mathbf{r}}{dt} = \dot r\hat{\mathbf{r}} + r\dot\theta\hat{\boldsymbol\theta},
\boldsymbol{a} = \frac{d\boldsymbol{v}}{dt} =\frac{d^2\mathbf{r}}{dt^2} = (\ddot r - r\dot\theta^2)\hat{\mathbf{r}} + (r\ddot\theta + 2\dot r \dot\theta)\hat{\boldsymbol\theta} \ ,

where dot-overmarkings indicate time differentiation. With this form for the acceleration \boldsymbol{a}, in an inertial frame of reference Newton's second law expressed in polar coordinates is:

\boldsymbol{F} = m \boldsymbol{a} = m(\ddot r - r\dot\theta^2)\hat{\mathbf{r}} + m(r\ddot\theta + 2\dot r \dot\theta)\hat{\boldsymbol\theta} \ ,

where F is the net real force on the particle. No fictitious forces appear because all fictitious forces are zero by definition in an inertial frame. However, from a mathematical standpoint it is handy to put only the second-order derivatives on the right side of this equation; that is we write the above equation by rearrangement of terms as:

\boldsymbol{F} +m r\dot\theta^2\hat{\mathbf{r}} - m 2\dot r \dot\theta\hat{\boldsymbol\theta} = m \tilde{\boldsymbol{a}}= m\ddot r \hat{\mathbf{r}} +m r\ddot\theta\hat{\boldsymbol\theta} \ ,

where a "coordinate" version of the "acceleration" is introduced:

\tilde{\boldsymbol{a}}= m\ddot r \hat{\mathbf{r}} +m r\ddot\theta\hat{\boldsymbol\theta} \ ,

consisting of only second-order time derivatives of the coordinates r and θ. The terms moved to the force-side of the equation are now treated as extra "fictitious forces" and, confusingly, are called the "centrifugal" and "Coriolis" force. However, these "forces" appear in an inertial frame, and so certainly are not the same as the fictitious forces that are zero in an inertial frame and non-zero only in a non-inertial frame.[59] In this article, these forces are called the "coordinate" centrifugal force and the "coordinate" Coriolis force to separate them from the "state-of-motion" forces.

Polar coordinates in a rotating frame of reference

Next, the same approach is used to find the fictitious forces of a (non-inertial) rotating frame. For example, if a rotating polar coordinate system is adopted for use in a rotating frame of observation, both rotating at the same counterclockwise rate Ω, we find the equations of motion in this frame as follows: the radial coordinate in the rotating frame is taken as r, but the angle θ' in the rotating frame changes with time:

\theta ' = \theta - \Omega t \ .

Consequently,

\dot\theta ' = \dot\theta - \Omega \ .

Plugging this result into the acceleration:

\frac{d^2\mathbf{r}}{dt^2} = \left( \ddot r - r \left( \dot\theta ' +\Omega\right) ^2 \right) \hat{\mathbf{r}} + \left( r\ddot\theta ' + 2\dot r \left(\dot\theta ' + \Omega \right) \right)\hat{\boldsymbol\theta}
=(\ddot r - r\dot\theta '^2)\hat{\mathbf{r}} + (r\ddot\theta' + 2\dot r \dot\theta ')\hat{\boldsymbol\theta} - \left( 2 r \Omega \dot\theta ' + r \Omega^2 \right)\hat{\mathbf{r}} + \left( 2 \dot r \Omega \right) \hat{\boldsymbol\theta}

The leading two terms are the same form as those in the inertial frame, and they are the only terms if the frame is not rotating, that is, if Ω=0. However, in this rotating frame we have the extra terms:

- \left( 2 r \Omega \dot\theta ' + r \Omega^2 \right)\hat{\mathbf{r}} + \left( 2 \dot r \Omega \right) \hat{\boldsymbol\theta}

The radial term Ω2 r is the centrifugal force per unit mass due to the system's rotation at rate Ω and the radial term 2 r \Omega \dot\theta ' is the radial component of the Coriolis force per unit mass, where r \dot\theta ' is the tangential component of the particle velocity as seen in the rotating frame. The term - \left( 2 \dot r \Omega \right) \hat{\boldsymbol\theta} is the so-called azimuthal component of the Coriolis force per unit mass. In fact, these extra terms can be used to measure Ω and provide a test to see whether or not the frame is rotating, just as explained in the example of rotating identical spheres. If the particle's motion can be described by the observer using Newton's laws of motion without these Ω-dependent terms, the observer is in an inertial frame of reference where Ω=0.

In this example we see that the "extra terms" in the acceleration of the particle are the same as the "state-of-motion" fictitious forces of the non-inertial rotating reference frame.[60]

What are the "coordinate" fictitious forces? As before, suppose we choose to put only the second-order time derivatives on the right side of Newton's law:

\boldsymbol{F} +m r\dot\theta '^2\hat{\mathbf{r}} -m 2\dot r \dot\theta '\hat{\boldsymbol\theta} +m \left( 2 r \Omega \dot\theta ' -m r \Omega^2 \right)\hat{\mathbf{r}} - m\left( 2 \dot r \Omega \right) \hat{\boldsymbol\theta} = m\ddot r\hat{\mathbf{r}}+ m r\ddot\theta'\ \hat{\boldsymbol\theta} = m\tilde{\boldsymbol{a} }

If we choose for convenience to treat \tilde{\boldsymbol{a}} as the so-called "acceleration", then the terms ( m r\dot\theta '^2\hat{\mathbf{r}} -m 2\dot r \dot\theta '\hat{\boldsymbol\theta}) are added to the so-called "fictitious force", which are not "state-of-motion" fictitious forces, but are actually components of force that persist even when Ω=0, that is, they persist even in an inertial frame of reference. Because these extra terms are added, the "coordinate" fictitious force is not the same as the "state-of-motion" fictitious force. Consequently, the "coordinate" fictitious force is not zero even in an inertial frame of reference.

General motion

When describing general motion in an inertial frame of reference, the actual forces acting on a particle often are referred to the instantaneous osculating circle tangent to the path of motion, and this circle in the general case is not centered at a fixed location. Consequently, description of the motion as though it were circular motion using a polar coordinate system with fixed origin is possible only for time intervals brief enough to approximate the motion with a fixed center of rotation. The osculating circle also is behind the non-inertial local coordinate system that is described in another subsection.

Centrifugal force in curvilinear coordinates

See also: Curvilinear coordinate system and Covariant derivative
Figure 11: Coordinate surfaces, coordinate lines, and coordinate axes of general curvilinear coordinates.
Figure 11: Coordinate surfaces, coordinate lines, and coordinate axes of general curvilinear coordinates.

Instead of Cartesian coordinates, when equations of motion are expressed in a curvilinear coordinate system, Christoffel symbols appear in the acceleration of a particle expressed in this coordinate system, as described below in more detail. Consider description of a particle motion from the viewpoint of an inertial frame of reference in curvilinear coordinates. Suppose the position of a point P in Cartesian coordinates is (x, y, z) and in curvilinear coordinates is (q1, q2. q3). Then functions exist that relate these descriptions:

x = x(q_1,\ q_2,\ q_3)\ ; \ q_1 = q_1(x,\ y, \ z) \ ,

and so forth. (The number of dimensions may be larger than three.) An important aspect of such coordinate systems is the element of arc length that allows distances to be determined. If the curvilinear coordinates form an orthogonal coordinate system, the element of arc length ds is expressed as:

ds^2 = \sum_{k=1}^{d} \left( h_{k}\right)^{2} \left( dq_{k} \right)^{2} \ ,

where the quantities hk are called scale factors.[61] A change dqk in qk causes a displacement hk dqk along the coordinate line for qk. At a point P, we place unit vectors ek each tangent to a coordinate line of a variable qk. Then any vector can be expressed in terms of these basis vectors, for example, from an inertial frame of reference, the position vector of a moving particle r located at time t at position P becomes:

\boldsymbol{r} =\sum_{k=1}^{d} q_k \ \boldsymbol{e_k} \,

where qk is the vector dot product of r and ek. The velocity v of a particle at P, can be expressed at P as:

\boldsymbol{v} =\sum_{k=1}^{d} v_k \ \boldsymbol{e_k} \,
=\frac{d}{dt}\boldsymbol {r} =\sum_{k=1}^{d} \dot q_k \ \boldsymbol{e_k} + \sum_{k=1}^{d} q_k \ \dot \boldsymbol{e_k} \,

where vk is the vector dot product of v and ek, and over dots indicate time differentiation. The time derivatives of the basis vectors can be expressed in terms of the scale factors introduced above. for example:

\frac{\partial}{\partial q_2} \boldsymbol {e_1} = -\boldsymbol{e}_2 \frac{1}{h_2}\frac{\partial h_1}{\partial q_2} -\boldsymbol{e}_3 \frac{1}{h_3}\frac{\partial h_1}{\partial q_3} \ ,  or, in general,   \frac { \partial \boldsymbol{e_j} } { \partial q_k} = \sum_{n=1}^{d} {\Gamma^n}_{kj}\boldsymbol{e_n} \ ,

in which the coefficients of the unit vectors are the Christoffel symbols for the coordinate system. The general notation and formulas for the Christoffel symbols are:[62][63]

{\Gamma^i}_{ii}=\begin{Bmatrix} \,i\,\\ i\,\,i \end{Bmatrix} = \frac{1}{h_i}\frac{\partial h_i}{\partial q_i}\! \ ;\  {\Gamma^i}_{ij}=\ \begin{Bmatrix} \,i\,\\ i\,\,j \end{Bmatrix} = \frac{1}{h_i}\frac{\partial h_i}{\partial q_j}= \begin{Bmatrix} \,i\,\\ j\,\,i \end{Bmatrix}\! \ ;\  {\Gamma^j}_{ii}=\begin{Bmatrix} \,j\,\\ i\,\,i \end{Bmatrix} = -\frac{h_i}{{h_j}^2}\frac{\partial h_i}{\partial q_j} \ ,

and the symbol is zero when all the indices are different. Using relations like this one,

\dot \boldsymbol{e_j} =\sum_{k=1}^{d}\frac {\partial}{\partial q_k}\boldsymbol{e_j}\dot q_k \
=\sum_{k=1}^{d} \sum_{i=1}^{d} {\Gamma^k}_{ij}\dot q_i \boldsymbol{e_k}\ ,

which allows all the time derivatives to be evaluated. For example, for the velocity:

\boldsymbol{v} =\frac{d}{dt}\boldsymbol {r} =\sum_{k=1}^{d} \dot q_k \ \boldsymbol{e_k} + \sum_{k=1}^{d} q_k \ \dot \boldsymbol{e_k}
=\sum_{k=1}^{d} \dot q_k \ \boldsymbol{e_k} + \sum_{j=1}^{d} q_j \ \dot \boldsymbol{e_j} ,
=\sum_{k=1}^{d} \dot q_k \ \boldsymbol{e_k} + \sum_{k=1}^{d}\sum_{j=1}^{d}\sum_{i=1}^{d} q_j \ {\Gamma^k}_{ij} \boldsymbol {e_k} \dot q_i \
=\sum_{k=1}^{d}\left( \dot q_k \ + \sum_{j=1}^{d}\sum_{i=1}^{d} q_j \ {\Gamma^k}_{ij} \dot q_i \right) \boldsymbol{e_k} \ ,

with the Γ-notation for the Christoffel symbols replacing the curly bracket notation. Using the same approach, the acceleration is then

\boldsymbol{a} = \frac{d}{dt} \boldsymbol{v} = \sum_{k=1}^{d} \dot v_k \ \boldsymbol{e_k} + \sum_{k=1}^{d} v_k \ \dot \boldsymbol{e_k} \ .
= \sum_{k=1}^{d} \left(\dot v_k \ + \sum_{j=1}^{d} \sum_{i=1}^{d}v_j{\Gamma^k}_{ij}\dot q_i \right)\boldsymbol{e_k} \ .

Looking at the relation for acceleration, the first summation contains the time derivatives of velocity, which would be associated with acceleration if these were Cartesian coordinates, and the second summation (the one with Christoffel symbols) contains terms related to the way the unit vectors change with time.[64]

"State-of-motion" versus "coordinate" fictitious forces

Earlier in this article a distinction was introduced between two terminologies, the fictitious forces that vanish in an inertial frame of reference are called in this article the "state-of-motion" fictitious forces and those that originate from differentiation in a particular coordinate system are called "coordinate" fictitious forces. Using the expression for the acceleration above, Newton's law of motion in the inertial frame of reference becomes:

\boldsymbol {F} =m\boldsymbol{a} =m \sum_{k=1}^{d} \left(\dot v_k \ + \sum_{j=1}^{d} \sum_{i=1}^{d}v_j{\Gamma^k}_{ij}\dot q_i \right)\boldsymbol{e_k} \ ,

where F is the net real force on the particle. No "state-of-motion" fictitious forces are present because the frame is inertial, and "state-of-motion" fictitious forces are zero in an inertial frame, by definition.

The "coordinate" approach to Newton's law above is to retain the second-order time derivatives of the coordinates {qk} as the only terms on the right side of this equation, motivated more by mathematical convenience than by physics. To that end, the force law can be re-written, taking the second summation to the force-side of the equation as:

\boldsymbol {F} -m \sum_{j=1}^{d} \sum_{i=1}^{d}v_j{\Gamma^k}_{ij}\dot q_i \boldsymbol{e_k} =m\tilde{\boldsymbol{a}}\ ,

with the convention that the "acceleration" \tilde{\boldsymbol{a}} is now:

\tilde{\boldsymbol{a}} = \sum_{k=1}^{d} \dot v_k\boldsymbol{e_k} \ .

In the expression above, the summation added to the force-side of the equation now is treated as if added forces were present. These summation terms are cust