Centrifugal force encyclopedia topics

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: 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. 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. Other examples are such systems as planets, centrifuges, carousels, turning cars, spinning buckets, and rotating space stations.

Choice of observational frame of reference

Centrifugal force can be confusing, which is nothing new to physics. 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, 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). 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 a rather more cerebral (but accurate) description of what the passenger feels as provided by Newton's laws in an inertial frame of reference; the passenger can imagine watching themselves in a kind of "out of body experience" or autoscopy. According to that description, 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. 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.

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, 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.

Are centrifugal and Coriolis forces "real"?

According to the first postulate of the special theory of relativity: which postulate allows identification of what is termed an inertial frame of reference. The centrifugal and Coriolis forces are called fictitious because they do not appear in an inertial frame of reference. 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.

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. However, apart from such fundamental issues of physics, for purposes of mechanics, 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. 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. Differences between artificial and real gravity can affect human health, and are a subject of study. 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, the simplest explanation is often to be preferred. The simplest explanation often involves fictitious forces.

Fictitious forces

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.

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":

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

where v_rot 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 × v_rot. To evaluate the Coriolis force, we need the velocity as seen in the rotating frame. Some pondering 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.

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

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

In discussion of a particle moving in a circular orbit, 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, first consider a coordinate system commonly used in planar motion, the so-called local coordinate system. See Figure 2.

Note: Observational frames of reference and coordinate systems are independent ideas: one is a concept of physics, the other of mathematics. An observational frame (such as an inertial frame or non-inertial frame of reference) refers to the state of motion of the observer. A coordinate system refers to a choice of language used to describe observations. Consequently, an observational frame of reference can choose to employ any coordinate system (polar, elliptical, curvilinear, …) to describe observations in that system. A change in this descriptive coordinate system does not change an observer's state of motion, and so does not entail a change in the observer's observational frame of reference, and vice versa. Below is a quotation concerning the relation between moving observational frames of reference and their associated coordinate systems:

and this:

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:

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. Then an incremental displacement along the path ds is described by:

dmathbf{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:

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{dtheta}{ds} .

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

frac{d sintheta}{ds} = cos theta frac {dtheta}{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)^2right)^{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 {dtheta}{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 .

With these results, 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 by introducing the rate at which the path is traveled, that is, by introducing the function s(t):

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 u_t(s) and u_n(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. 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

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):

$mathbf{a}_mathrm{rot},$	$=mathbf{a} - 2mathbf{Omega times v_mathrm{rot}} - mathbf{Omega times (Omega times r)} ,$
	$=mathbf{a + a_mathrm{coriolis} + a_mathrm{centrifugal}} ,$ ,

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, $mathbf{v_mathrm{rot}},$ 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

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, a third term, the Euler acceleration must be added if the rotation of the frame varies, and a fourth acceleration is needed if the frame is linearly accelerating.

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 3 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 4 shows two identical spheres rotating about the center of the string joining them. This sphere example is one used by Newton himself. 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 5. These two forces are provided by the string, putting the string under tension, also shown in Figure 5.

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{u}_R  ,

where u_R 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, |x_B| = 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(omega R right) mathbf{u}_R ,

where R is the distance to the object from the center of rotation, and v_B is the velocity of the object subject to the Coriolis force, |v_B| = 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 ω_I (I = inertial), and the frame rotates at a different rate ω_R (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 ω_S = ω_I − ω_R (S = spheres). That is, if the frame rotates more slowly than the spheres, ω_S > 0 and the spheres advance counterclockwise around a circle, while for a more rapidly moving frame, ω_S < 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 ω_I and ω_S is greater. The reason for the sign change is that when ω_I > ω_S, 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 ω_I < ω_S, 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 F_Fict 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 F_Fict 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{d boldsymbolOmega  }{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 = Rmathbf{u}_R  ,

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

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

where u_θ is a unit vector perpendicular to u_R pointing in the direction of motion.

The vector of rotation Ω = ω_R u_z (u_z a unit vector in the z-direction), and Ω × u_R = ω_R (u_z × u_R) = ω_R u_θ ; Ω × u_θ = −ω_R u_R. The centrifugal force is then:

mathbf{F}_mathrm{Cfgl} = - m boldsymbolOmega  times (boldsymbolOmega times mathbf{x}_B ) =momega_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 boldsymbolOmega  times mathbf{v}_{B} = 2momega_S omega_R R mathbf{u}_R

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

mathbf{F}_{mathrm{Fict}} = mathbf{F}_mathrm{Cfgl} + mathbf{F}_mathrm{Cor}

=left(momega_R^2 R + 2momega_S omega_R Rright) mathbf{u}_R = momega_R left(omega_R + 2omega_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^2right) 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 ω_I, ω_S is the greater, inasmuch as the centrifugal force contribution always is outward.

Dropping ball

Figure 6 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 6 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 6. 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 6 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 7.

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 6 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 8 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 8. The stationary observer sees a uniform circular motion of the parachutist when the motion is projected downward, as in the left panel of Figure 8. 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 8, 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.

Potential energy

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:

F_{Cfgl} = -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 9.

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.

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 Principia:

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.

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.

External links

Motion over a flat surface Java physlet by Brian Fiedler (from School of Meteorology at the University of Oklahoma) illustrating fictitious forces. The physlet shows both the perspective as seen from a rotating and from a non-rotating point of view.
Motion over a parabolic surface Java physlet by Brian Fiedler (from School of Meteorology at the University of Oklahoma) illustrating fictitious forces. The physlet shows both the perspective as seen from a rotating and as seen from a non-rotating point of view.
Animation clip showing scenes as viewed from both an inertial frame and a rotating frame of reference, visualizing the Coriolis and centrifugal forces.
Centripetal and Centrifugal Forces at MathPages
Centrifugal Force at h2g2
What is a centrifuge?
John Baez: Does centrifugal force hold the Moon up?
XKCD demonstrates the life and death importance of centrifugal force

Notes and references

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Wikipedia

In classical mechanics, centrifugal force (from Latin centrum "center" and fugere "to flee") is an apparent force acting outward from the axis of rotation of a rotating reference frame.

Analysis using fictitious forces

Advantages of rotating frames

Choice of observational frame of reference

Are centrifugal and Coriolis forces "real"?

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

Fictitious forces

$mathbf{F}_mathrm{centrifugal} ,$	$= m mathbf{a}_mathrm{centrifugal} ,$
	$=m omega^2 mathbf{R} ,$

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

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

Moving objects and observational frames of reference

and this:

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

dmathbf{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:

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.

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{dtheta}{ds} .

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

frac{d sintheta}{ds} = cos theta frac {dtheta}{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)^2right)^{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 {dtheta}{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 .

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)  ,

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

Uniformly rotating reference frames

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

$mathbf{a}_mathrm{rot},$	$=mathbf{a} - 2mathbf{Omega times v_mathrm{rot}} - mathbf{Omega times (Omega times r)} ,$
	$=mathbf{a + a_mathrm{coriolis} + a_mathrm{centrifugal}} ,$ ,

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.

Analysis using fictitious forces

Advantages of rotating frames

Choice of observational frame of reference

Are centrifugal and Coriolis forces "real"?

Fictitious forces

Moving objects and observational frames of reference

Uniformly rotating reference frames

Non uniformly rotating reference frame

Examples

Whirling table

Rotating identical spheres

Inertial frame

Rotating frame

Coriolis force

General case

Is the fictitious force ad hoc?

Dropping ball

Inertial frame

Uniformly rotating frame

Parachutist

Potential energy

Development of the modern conception of centrifugal force

Applications

Further reading

External links

See also

Notes and references

Analysis using fictitious forces

Advantages of rotating frames

Choice of observational frame of reference

Are centrifugal and Coriolis forces "real"?

Fictitious forces

Moving objects and observational frames of reference

Uniformly rotating reference frames

Non uniformly rotating reference frame