Centrifugal force

From Wikipedia, the free encyclopedia

For the real outward-acting force that can be found in circular motion, see Reactive centrifugal force.
For the external force required to make a body follow a curved path, see Centripetal force.

In classical mechanics, centrifugal force (from Latin centrum "center" and fugere "to flee") is an apparent force acting outward from the axis of a rotating reference frame.[1][2] Centrifugal force is a fictitious force[3][4] (also known as a pseudo force, inertial force or d'Alembert force) meaning that it is an artifact of acceleration of a reference frame.[5][6][7] Unlike real forces such as gravitational or electromagnetic forces, fictitious forces do not originate from physical interactions between objects, and they do not appear in Newton's laws of motion for 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, however, fictitious forces must be included along with the real forces in order to make accurate physical predictions. The fictitious forces present in a rotating reference frame with a uniform angular velocity are the centrifugal force and the Coriolis force, to which is added the Euler force when angular velocity is time dependent.[8]

In certain situations a rotating reference frame has advantages over an inertial reference frame.[9][10] For example, a rotating frame of reference is more convenient for description of what happens on the inside of a car going around a corner, or inside a centrifuge, or in the artificial gravity of a rotating space station. Centrifugal force is used in the FAA pilot's manual in describing turns.[11] Centrifugal force and other fictitious forces can be used to think about these systems, and calculate motions within them. With the addition of fictitious forces, Newton's laws can be used in non-inertial reference frames such as planets, centrifuges, carousels, turning cars, and spinning buckets,[12] though the fictitious forces themselves do not obey Newton's third law.[13]

As discussed in detail below, within a rotating frame, centrifugal force acts on anything with mass, depends only on the position and the mass of the object, and always is oriented outward from the axis of rotation of the rotating frame. The Coriolis force depends on both the velocity and mass of the object, but is independent of its position.[9][10]

Contents

[edit] Choice of frame of reference

The Newtonian concept of force had to overcome numerous "common sense" perceptions, among them the perception of friction as a property of motion rather than a force,[14][15] and the failure to recognize change of direction as of equal importance to change in speed (that is the concept of velocity as a vector quantity).[16] A confusing concept related to change of direction is centrifugal force,[17] which often is experienced as a force, and indeed provides a natural explanation of some problems involving rotation. However, our experience (for example, as inhabitants of the Earth, or passengers in a turning car) is seen from our rotating reference frame, which is not the reference frame in which Newton's law of inertia applies (the inertial reference frame). In our rotating frame, centrifugal force pushes us away from the center of rotation; but from the view of an inertial frame, it is the tendency of all bodies to maintain velocity in a constant direction that leads us to experience a centrifugal force. To elaborate, a body in circular motion at each instant tends to move in a straight line tangent to the circular orbit, and so appears to be moving away from the center of rotation: it "pushes away". To the inertial observer viewing matters with Newton's laws, the body simply is following the law of inertia, and therefore defying the attempt to make it follow a circular path: to constrain the body to the circular path, centripetal force must be exerted.

A very common experience is that of being pushed against the door of a turning car. Our experience is the centrifugal force. A rather more cerebral (but accurate) description of what we feel is to use Newton's laws in an inertial frame of reference, watching ourselves in an "out of body experience". That description says our body tends to travel in a straight line, but the car is going in a circle. Therefore, the car pushes us to keep us turning, not the other way around (we are pushing on the car door, but it is a reaction to the car pushing on us). Further discussion of this example can be found in the article on reactive centrifugal force.

Here is a related example that illustrates 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.[18] 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.[19][20]

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[17][21], 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.[17][21][22]

[edit] Are centrifugal and Coriolis forces "real"?

See also: Gravitron

The centrifugal and Coriolis forces are called fictitious because they do not appear in an inertial frame of reference.[23] Despite the name, fictitious forces are experienced as very real to those actually in a non-inertial frame. Fictitious forces also provide a convenient way to discuss dynamics within rotating environments, and can simplify explanations and mathematics.[9][24]

An interesting exploration of the reality of centrifugal force is provided by artificial gravity introduced into a space station by rotation.[25] 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.[26][27][28] These differences between artificial and real gravity can affect human health, and are a subject of study.[29] In any event, the "fictitious" forces in this habitat would seem perfectly real to those living in the station. Although they could readily do experiments that would reveal the space station was rotating, inhabitants would find description of daily life remained more natural in terms of fictitious forces, as discussed next.

[edit] Fictitious forces

Main article: Fictitious force

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.[8] In particular, 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}_\perp \,

where m\, is the mass of the object, and r\perp is the vector that locates the object relative to the center of rotation (r\perp 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":[30]

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

where vrot is the velocity as seen in the rotating frame of reference.

Here is an example. A body that is stationary relative to the non-rotating inertial frame will be rotating when viewed from the rotating frame. Therefore, Newton's laws, as applied to what looks like circular motion in the rotating frame, requires an inward centripetal force of −m ω2 r\perp 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\perp and the Coriolis force −2m Ω × v.[31] [32] 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\perp. The combination of the centrifugal and Coriolis force is then m ω2 r\perp−2m ω2 r\perp = −m ω2 r\perp, exactly the centripetal force required by Newton's laws for circular motion.[33] [34][35]

For further examples and discussion, see below, and see Taylor.[36]

Because this centripetal force is combined from only pseudo forces, it is "fictitious" in the sense of having no apparent origin in physical sources (like charges or gravitational bodies); and having no apparent source, it is simply 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.

[edit] Moving objects and frames of reference

See also: Generalized forces, Generalized force, Curvilinear coordinates, Generalized coordinates, and Frenet-Serret formulas

In discussion of an object 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 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.

And what is the parallel in the case of an elliptical orbit? Suppose we identify the forces normal to the trajectory as centripetal forces and those parallel to the trajectory as "tangential" forces. What switch of hats leads to fictitious centrifugal and Euler forces? Apparently one must switch to a continuously changing frame of reference, whose origin 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 object about that origin at time t. For that to make sense, one has to sit on the object, with a local coordinate system that has unit vectors normal to the trajectory and parallel to it. So, for a pilot in an airplane, the fictitious forces are a matter of direct experience, but they cannot be related to a simple observational frame of reference other than the airplane itself unless the airplane 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 that limited time, the centrifugal and Euler forces can be analyzed on the basis of circular motion. See article discussing turning an airplane.

Next, rotating reference frames are discussed in more detail.

[edit] Uniformly rotating reference frames

See also: Circular motion and Uniform circular motion

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:[23][37][10]

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

where \mathbf{a}_\mathrm{rot}\, is the acceleration relative to the rotating frame, \mathbf{a}\, is the acceleration relative to the inertial frame, \mathbf{\Omega}\, is the angular velocity vector describing the rotation of the reference frame,[38] \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}_\perp ,

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

[edit] 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,[9][10] a third term, the Euler acceleration must be added if the rotation of the frame varies,[39] and a fourth acceleration is needed if the frame is linearly accelerating.[40]

[edit] Examples

See also: Fictitious force#Crossing a carousel

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.

[edit] ♦ Whirling table

Figure 1: 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 1: 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 1 shows a simplified version of an apparatus for studying centrifugal force called the "whirling table".[41] 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.

[edit] ♦ Rotating identical spheres

Figure 2: 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 2: 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 3: 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 3: 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 2 shows two identical spheres rotating about the center of the string joining them. This example is one used by Newton himself.[42] 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.

[edit] 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 3. These two forces are provided by the string, putting the string under tension, also shown in Figure 3.

[edit] 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.

[edit] 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:[31]

\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:[31]

\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.

[edit] 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 ). 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.)

[edit] 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 whether this solution fits in with general experience with other situations, or is 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):[43]

\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 \ .

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 = ωRk (k a unit vector in the z-direction), and Ω × 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 ).[44] Combining the terms:

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

Consequently, the fictitious force found above for this problem of rotating spheres is consistent with the general result and is not just an ad hoc solution "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.

[edit] ♦ Dropping ball

Figure 4: 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 4: 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 5: 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 related to the negative of the Coriolis force (the Coriolis force is −2 m Ω × v) as found using the right-hand rule for vector cross products.
Figure 5: 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 related to the negative of the Coriolis force (the Coriolis force is −2 m Ω × v) as found using the right-hand rule for vector cross products.

Figure 4 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 3 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.

[edit] 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.

[edit] 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 4. 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 3 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 5.

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.

[edit] ♦ Parachutist

Figure 6: 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 6: 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 4 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 6 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 6. The stationary observer sees a uniform circular motion of the parachutist when the motion is projected downward, as in the left panel of Figure 6. 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 6, 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. 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.

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

[edit] Potential energy

Figure 7: The interface of two immiscible liquids rotating around a vertical axis is an upward-opening circular paraboloid.
Figure 7: The interface of two immiscible liquids rotating around a vertical axis is an upward-opening circular paraboloid.

The fictitious centrifugal force is conservative and has a potential energy of the form

E_p = -\frac{1}{2} m \omega^2 r_\perp^2

This is useful, for example, in calculating the form of the water surface h(r)\, in a rotating bucket: requiring the potential energy per unit mass on the surface gh(r) - \frac{1}{2}\omega^2 r^2\, to be constant, we obtain the parabolic form h(r) = \frac{\omega^2}{2g}r^2 + C (where C is a constant). See Figure 7.

Similarly, the potential energy of the centrifugal force is often used in the 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.

The Coriolis force has no equivalent potential, as it acts perpendicular to the velocity vector and hence rotates the direction of motion, but does not change the energy of a body.

[edit] Development of the modern conception of centrifugal force

 This article or section is in need of attention from an expert on the subject.

WikiProject History of Science or the History of Science Portal may be able to help recruit one.

If a more appropriate WikiProject or portal exists, please adjust this template accordingly.

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."[45]

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. [46]

The modern conception of centrifugal force appears to have its origins in Christiaan Huygens' paper De Vi Centrifuga, written in 1659.[47] It has been suggested that the idea of circular motion as caused by a single force was introduced to Newton by Robert Hooke.[46]

[edit] 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.

[edit] See also

Look up centrifugal force in
Wiktionary, the free dictionary.
Look up centrifugal in
Wiktionary, the free dictionary.

[edit] Notes and references

  1. ^ David McNaughton. Centrifugal and Coriolis Effects. Retrieved on 2008-05-18.
  2. ^ Lynda Williams. Centrifugal Effect. Retrieved on 2008-05-18.
  3. ^ Centrifugal Force. scienceworld.wolfram.com. Retrieved on 2008-05-18.
  4. ^ Centrifugal Force - Britannica online encyclopedia. Retrieved on 2008-05-18.
  5. ^ Max Born & Günther Leibfried (1962). Einstein's Theory of Relativity. New York: Courier Dover Publications, pp.76-78. ISBN 0486607690. 
  6. ^ NASA: Accelerated Frames of Reference: Inertial Forces
  7. ^ Science Joy Wagon: Centrifugal force - the false force
  8. ^ 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. 
  9. ^ a b c d John Robert Taylor (2004). Classical Mechanics. Sausalito CA: University Science Books, Chapter 9, pp. 327 ff. ISBN 189138922X. 
  10. ^ 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. 
  11. ^ Federal Aviation Administration (2007). Pilot's Encyclopedia of Aeronautical Knowledge. Oklahoma City OK: Skyhorse Publishing Inc., Figure 3-21. ISBN 1602390347. 
  12. ^ Louis N. Hand & Janet D. Finch (1998). Analytical Mechanics. Cambridge UK: Cambridge University Press, pp. 266-267. ISBN 0521575729. 
  13. ^ Because fictitious forces acting on an object do not originate from the action of other objects, there is nothing to experience an associated reaction force: fictitious forces do not obey Newton's third law.
  14. ^ Roger G. Newton (2007). From Clockwork to Crapshoot: A History of Physics. Cambridge MA: Belknap Press of Harvard University Press, p. 27. ISBN 0674023374. 
  15. ^ 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. 
  16. ^ Val Dusek (1999). The Holistic Inspirations of Physics: The Underground History of Electromagnetic Theory. New Brunswick NJ: Rutgers University Press, p. 187. ISBN 0813526353. 
  17. ^ a b c "Force Concept Inventory" . Physics Teacher, v30 n3 p141-58 Mar 1992. 
  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. ^ a b John Robert Taylor (2004). Classical Mechanics. Sausalito CA: University Science Books, pp. 343-344. ISBN 189138922X. 
  24. ^ G. K. Batchelor (2000). An Introduction to Fluid Dynamics. Cambridge UK: Cambridge University Press, pp.138-139. ISBN 0521663962. 
  25. ^ The pull of hypergravity
  26. ^ Sheldon Marshall Ebenholtz (2001). Oculomotor Systems and Perception. Cambridge UK: Cambridge University Press, pp. 151-153. ISBN 0521804590. 
  27. ^ For more detail see Hall: Artificial gravity and the architecture of orbital habitats.
  28. ^ Hall: Inhabiting artificial gravity
  29. ^ See, for example, Pouly and Young.
  30. ^ Henry M. Stommel & Dennis W. Moore (1989). An Introduction to the Coriolis Force. Columbia University Press, p. 12. ISBN 0231066368. 
  31. ^ a b c Georg Joos & Ira M. Freeman (1986). Theoretical Physics. New York: Courier Dover Publications, p. 233. ISBN 0486652270. 
  32. ^ John Robert Taylor (2004). Classical Mechanics. Sausalito CA: University Science Books, pp. 348-349. ISBN 189138922X. 
  33. ^ Louis Bevier Spinney (1911). A Text-book of Physics. Macmillan Co., pp. 47-49. 
  34. ^ 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. 
  35. ^ Burgel, B. (1967). "Centrifugal Force". American Journal of Physics 35: 649. 
  36. ^ John Robert Taylor (2004). pp. 349ff. ISBN 189138922X. 
  37. ^ Vladimir Igorevich Arnolʹd (1989). Mathematical Methods of Classical Mechanics. Berlin: Springer, §27 pp. 129 ff.. ISBN 0387968903. 
  38. ^ 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.
  39. ^ Jerrold E. Marsden, Tudor S. Ratiu (1999). Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems. Springer, p. 251. ISBN 038798643X. 
  40. ^ Vladimir Igorevich Arnolʹd (1989). Mathematical Methods of Classical Mechanics. Berlin: Springer, §27 pp. 129 ff.. ISBN 0387968903. 
  41. ^ Dionysius Lardner (1877). Mechanics. Oxford University Press, p. 150. 
  42. ^ Max Born (1962). Einstein's Theory of Relativity. Courier Dover Publications, Figure 43, p. 79. ISBN 0486607690. 
  43. ^ 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. 
  44. ^ The case ωS < 0 applies to the above example with spheres at rest in the inertial frame.
  45. ^ 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.
  46. ^ a b Newton, Sir Isaac. Retrieved on 2008-05-25.
  47. ^ Soshichi Uchii (October 9, 2001). Inertia. Retrieved on 2008-05-25.

[edit] Further reading

[edit] External links