Centrifugal force (rotating reference frame)

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  (Redirected from Centrifugal force) 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. 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.

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.

Although frames rotating about a fixed axis are the most commonly analyzed, centrifugal forces also arise in more general non-inertial reference frames. See, for example, Centrifugal force (planar motion). This article focuses upon the cases of rotation about a fixed axis.

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:

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. 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. 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 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. 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"?

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; that is, neither rotating nor accelerating relative to the stars. 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:

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. 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 discussion of the reality of fictitious forces is provided by Kompaneyets:

"Naturally, the acceleration of a point caused by noninertiality of the system is absolutely real, relative to that system, in spite of the fact that there are other, inertial, systems relative to which this acceleration does not exist. In [the equation for acceleration] this acceleration is written as if it were due to some additional forces. These forces are usually called inertial forces. In so far as the acceleration associated with them is in every way real, the discussion (which sometimes arises) about the reality of inertial forces themselves must be considered as aimless. It is only possible to talk about the difference between the forces of inertia and the forces of interaction between bodies."

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

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.

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} \,
m\,

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 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. 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.

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):

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

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, 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.

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 contour lines 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 contour lines 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 contour lines 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 contour lines 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:

= -\frac{\partial }{\partial r} E_p
h(r)\,
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". 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. 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:


\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:


\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):


\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 ). Combining the terms:


<img  alt=
=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.

Development of the modern conception of centrifugal force

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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:

COR. Since the centrifugal force of the parts of the earth, arising from the earth's diurnal motion, which is to the force of gravity as 1 to 289, raises the waters under the equator to a height exceeding that under the poles by 85472 Paris feet, as above, in Prop. XIX., the force of the sun, which we have now shewed to be to the force of gravity as 1 to 12868200, and therefore is to that centrifugal force as 289 to 12868200, or as 1 to 44527, will be able to raise the waters in the places directly under and directly opposed to the sun to a height exceeding that in the places which are 90 degrees removed from the sun only by one Paris foot and 113 V inches ; for this measure is to the measure of 85472 feet as 1 to 44527.

Newton: Principia Corollary to Book II, Proposition XXXVI. Problem XVII

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.

Notes and references

  1. ^ Fictitious forces (also known as a pseudo forces, inertial forces or d'Alembert forces), exist for observers in a rotating reference frame. See, for example, "Centrifugal Force". scienceworld.wolfram.com. Retrieved on 2008-05-18., "Centrifugal Force - Britannica online encyclopedia". Retrieved on 2008-05-18., Max Born & Günther Leibfried (1962). Einstein's Theory of Relativity. New York: Courier Dover Publications, pp.76-78. ISBN 0486607690. , NASA: Accelerated Frames of Reference: Inertial Forces, Science Joy Wagon: Centrifugal force - the false force
  2. ^ a b Jerrold E. Marsden, Tudor S. Ratiu (1999). Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems. Springer, p. 251. ISBN 038798643X. 
  3. ^ a b c John Robert Taylor (2004). Classical Mechanics. Sausalito CA: University Science Books, Chapter 9, pp. 327 ff. ISBN 189138922X. 
  4. ^ a b c d Stephen T. Thornton & Jerry B. Marion (2004). Classical Dynamics of Particles and Systems, 5th Edition, Belmont CA: Brook/Cole, Chapter 10. ISBN 0534408966. 
  5. ^ David McNaughton. "Centrifugal and Coriolis Effects". Retrieved on 2008-05-18.
  6. ^ Lynda Williams. "Centrifugal Effect". Retrieved on 2008-05-18.
  7. ^ Cases in which the axis of rotation itself changes direction in time, as in rigid body motion, can be treated using Euler angles.
  8. ^ K.S. Rao (2003). Classical Mechanics. Orient Longman, p. 162. ISBN 8173714363. 
  9. ^ Louis N. Hand, Janet D. Finch (1998). Analytical Mechanics. Cambridge University Press, p. 267. ISBN 0521575729. 
  10. ^ Federal Aviation Administration (2007). Pilot's Encyclopedia of Aeronautical Knowledge. Oklahoma City OK: Skyhorse Publishing Inc., Figure 3-21. ISBN 1602390347. 
  11. ^ Richard Hubbard (2000). Boater's Bowditch: The Small Craft American Practical Navigator. NY: McGraw-Hill Professional, p. 54. ISBN 0071361367. 
  12. ^ Lawrence K. Wang, Norman C. Pereira (1979). Handbook of Environmental Engineering: Air and Noise Pollution Control. Humana Press, p. 63. ISBN 0896030016. 
  13. ^ Lee M. Grenci, Jon M. Nese (2001). A World of Weather: Fundamentals of Meteorology. Kendall Hunt, p. 272. ISBN 0787277169. 
  14. ^ a b c David Hestenes, Malcolm Wells, and Gregg Swackhamer. "Force Concept Inventory". Physics Teacher, v30 n3 p141-58 Mar 1992. 
  15. ^ Roger G. Newton (2007). From Clockwork to Crapshoot: A History of Physics. Cambridge MA: Belknap Press of Harvard University Press, p. 27. ISBN 0674023374. 
  16. ^ Michael R. Matthews (2000). Time for science education: how teaching the history and philosophy of pendulum motion can contribute to science literacy. New York: Springer, p. 84. ISBN 0306458802. 
  17. ^ Val Dusek (1999). The Holistic Inspirations of Physics: The Underground History of Electromagnetic Theory. New Brunswick NJ: Rutgers University Press, p. 187. ISBN 0813526353. 
  18. ^ Louis Bevier Spinney (1911). §47: The cream separator and centrifugal drier, p. 52. 
  19. ^ SI Rubinow (2002 (1975)). Introduction to mathematical biology. Courier/Dover Publications, pp. 235-244. ISBN 0486425320. 
  20. ^ Jagannath Mazumdar (1999). An Introduction to Mathematical Physiology and Biology. Cambridge UK: Cambridge University Press, pp. 33 ff. ISBN 0521646758. 
  21. ^ a b R. Ploetzner and K. Van Lehn. "The Acquisition of Qualitative Physics Knowledge during Textbook-Based Physics Training".
  22. ^ Tom Henderson. "The Forbidden F-Word".
  23. ^ The question of "moving uniformly relative to what?" was answered by Newton as "relative to absolute space". As a practical matter, "absolute space" was considered to be the fixed stars. For a discussion of the role of fixed stars, see Henning Genz (2001). Nothingness: The Science of Empty Space. Da Capo Press, p. 150. ISBN 0738206105. 
  24. ^ Robert Resnick, David Halliday, Kenneth S. Krane (2001). Physics, 5th Edition, Wiley, Volume 1, Chapter 3. ISBN 0471320579. 
  25. ^ RG Takwale (1980). Introduction to classical mechanics. New Delhi: Tata McGraw-Hill, p. 70. ISBN 0070966176. 
  26. ^ NMJ Woodhouse (2003). Special relativity. London/Berlin: Springer, p. 6. ISBN 1-85233-426-6. 
  27. ^ a b John Robert Taylor (2004). Classical Mechanics. Sausalito CA: University Science Books, pp. 343-344. ISBN 189138922X. 
  28. ^ Kompaneet︠s︡, A. S. & George Yankovsky (2003). Theoretical Physics. Courier Dover Publications, p. 71. ISBN 0486495329. 
  29. ^ The pull of hypergravity
  30. ^ For more detail see Hall: Artificial gravity and the architecture of orbital habitats.
  31. ^ Hall: Inhabiting artificial gravity
  32. ^ See, for example, Pouly and Young.
  33. ^ Einstein wrote: "Things should be made as simple as possible, but not any simpler" Scott Thorpe (2000). How to think like Einstein: Simple ways to break the rules and discover your hidden genius. Sourcebooks, p. 35. ISBN 1570715858. 
  34. ^ Louis N. Hand & Janet D. Finch (1998). Finch Analytical Mechanics. Cambridge UK: Cambridge University Press, p. 267. ISBN 0521575729. 
  35. ^ John Robert Taylor (2004). Classical Mechanics. Sausalito CA: University Science Books, Chapter 9, p. 329. ISBN 189138922X. 
  36. ^ Henry M. Stommel & Dennis W. Moore (1989). An Introduction to the Coriolis Force. Columbia University Press, p. 12. ISBN 0231066368. 
  37. ^ a b c Georg Joos & Ira M. Freeman (1986). Theoretical Physics. New York: Courier Dover Publications, p. 233. ISBN 0486652270. 
  38. ^ John Robert Taylor (2004). Classical Mechanics. Sausalito CA: University Science Books, pp. 348-349. ISBN 189138922X. 
  39. ^ The vector cross product of the two orthogonal vectors Ω and R is a vector of magnitude equal to the product of their magnitudes, namely ΩR = ωR = vrot, and with direction given by the right-hand rule, in this case found by aligning the thumb with Ω, the index finger with R, and the middle finger normal to these two fingers points in the direction of −vrot.
  40. ^ Louis Bevier Spinney (1911). A Text-book of Physics. Macmillan Co., pp. 47-49. 
  41. ^ Arthur Beiser & George J. Hademenos (2003). Applied physics: Based on Schaum's Outline of Theory and Problems of Applied Physics (Third Edition). McGraw-Hill Professional, p. 37. ISBN 0071398783. 
  42. ^ Burgel, B. (1967). "Centrifugal Force". American Journal of Physics 35: 649. doi:10.1119/1.1974204. 
  43. ^ John Robert Taylor (2004). pp. 349ff. ISBN 189138922X. 
  44. ^ Vladimir Igorevich Arnolʹd (1989). Mathematical Methods of Classical Mechanics. Berlin: Springer, §27 pp. 129 ff.. ISBN 0387968903. 
  45. ^ This vector points along the axis of rotation with polarity determined by the right-hand rule and a magnitude |Ω| = ω = angular rate of rotation.
  46. ^ Jerrold E. Marsden, Tudor S. Ratiu (1999). Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems. Springer, p. 251. ISBN 038798643X. 
  47. ^ Vladimir Igorevich Arnolʹd (1989). Mathematical Methods of Classical Mechanics. Berlin: Springer, §27 pp. 129 ff.. ISBN 0387968903. 
  48. ^ Robert Daniel Carmichael (1920). The Theory of Relativity. John Wiley & Sons, p. 78. 
  49. ^ Dionysius Lardner (1877). Mechanics. Oxford University Press, p. 150. 
  50. ^ Max Born (1962). Einstein's Theory of Relativity. Courier Dover Publications, Figure 43, p. 79. ISBN 0486607690. 
  51. ^ Many sources are cited in Fictitious force. Here are two more: PF Srivastava (2007). Mechanics. New Delhi: New Age International Publishers, p. 43. ISBN 978-81-224-1905-4.  and NC Rana and PS Joag (2004). Mechanics. New Delhi: Tata McGraw-Hill, p. 99ff. ISBN 0074603159. 
  52. ^ The case ωS < 0 applies to the earlier example with spheres at rest in the inertial frame.
  53. ^ The Relativization of Centrifugal Force Author(s): Domenico Bertoloni Meli Source: Isis, Vol. 81, No. 1, (Mar., 1990), pp. 23-43 Published by: The University of Chicago Press on behalf of The History of Science Society.
  54. ^ a b "Newton, Sir Isaac". Retrieved on 2008-05-25.
  55. ^ Soshichi Uchii (October 9, 2001). "Inertia". Retrieved on 2008-05-25.

Further reading