• Czech: odstředivý
• Finnish: keskipakoinen, sentrifugaalinen
• French: centrifuge m|f

References

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. Centrifugal force is a fictitious force (also known as a pseudo force, inertial force or d'Alembert force) meaning that it is an artifact of acceleration of a reference frame. 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.

In certain situations a rotating reference frame has advantages over an inertial reference frame. 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. 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, though the fictitious forces themselves do not obey Newton's third law.

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. 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). A confusing concept related to change of direction is centrifugal force, 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. 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"?

seealso Gravitron The centrifugal and Coriolis forces are called fictitious because they do not appear in an inertial frame of reference. 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.

An interesting exploration of the reality of centrifugal force is provided by 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. These 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 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.

Fictitious forces

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

\mathbf_\mathrm = -2 \, m \, \boldsymbol \times \boldsymbol v_\ ,

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. 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.

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 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.

Moving objects and frames of reference

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.

Uniformly rotating reference frames

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

The laws of physics are the same in all inertial frames. But a rotating reference frame is not an inertial frame, so the laws of physics are transformed from the inertial frame to the rotating frame. For example, assuming a constant rotation speed, transformation is achieved by adding to every object two coordinate accelerations that correct for the constant rotation of the coordinate axes. The vector equations describing these accelerations are: \mathbf\, is the velocity of the body relative to the rotating frame, and \mathbf\, is the position vector of the body. The last term is the centrifugal acceleration:

\mathbf_\textrm = - \mathbf = \omega^2 \mathbf_\perp ,

where \mathbf is the component of \mathbf\, perpendicular to the axis of rotation.

Non uniformly rotating reference frame

Although changing coordinates from an inertial frame of reference to any rotating one alters the equations of motion to require the inclusion of two sources of fictitious force, the centrifugal force, and the Coriolis force, and a fourth acceleration is needed if the frame is linearly accelerating.

Examples

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

♦ Whirling table

Figure 1 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.

♦ Rotating identical spheres

Figure 2 shows two identical spheres rotating about the center of the string joining them. 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 3. These two forces are provided by the string, putting the string under tension, also shown in Figure 3.

Rotating frame

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

Coriolis force

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

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.

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.

See also

• Circular motion
• Coriolis force
• Centripetal force
• Equivalence principle
• Euler force - a force that appears when the frame angular rotation rate varies
• Folk physics
• Rotational motion
• Reactive centrifugal force - a force that occurs as reaction due to a centripetal force
• Lamm equation
• Orthogonal coordinates
• Frenet-Serret formulas
• Statics
• Kinetics (physics)
• Kinematics
• Applied mechanics
• Analytical mechanics
• Dynamics (physics)
• Classical mechanics

Notes and references

Further reading

• Newton's description in Principia
• Centrifugal reaction force - Columbia electronic encyclopedia
• M. Alonso and E.J. Finn, Fundamental university physics, Addison-Wesley
• Centripetal force vs. Centrifugal force - from an online Regents Exam physics tutorial by the Oswego City School District
• Centrifugal force acts inwards near a black hole
• Centrifugal force at the HyperPhysics concepts site
• A list of interesting links

External links

• Animation clip showing scenes as viewed from both an inertial frame and a rotating frame of reference, visualizing the Coriolis and centrifugal forces.
• Centripetal and Centrifugal Forces at MathPages
• Centrifugal Force at h2g2
• What is a centrifuge?
• John Baez: Does centrifugal force hold the Moon up?
• XKCD demonstrates the life and death importance of centrifugal force