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Centrifugal force (rotating reference frame)

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It has been suggested that Reactive centrifugal force be merged into this article. (Discuss) Proposed since May 2008.
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 physics, centrifugal force (from Latin centrum "center" and fugere "to flee") is a fictitious force that appears when describing physics in a rotating reference frame; it acts on anything with mass considered in such a frame. Centrifugal force is considered to be a fictitious force (or pseudo force), because it does not appear when the motion is expressed in an inertial frame of reference, in which the motion of an object is explained by the real impressed forces, together with inertia.

Changing coordinates from an inertial frame of reference to a rotating one alters the equations of motion to include two sources of fictitious force to compensate for the non-inertial character of the frame. The centrifugal force depends only on the position and the mass of the object and is always oriented away from the axis of rotation of the rotating frame, whereas the Coriolis force depends on the velocity and mass of the object but is independent of its position.

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.

Colloquially, the term "centrifugal force" is sometimes also used to refer to any force pushing away from a center in connection with rotation; this article discusses only the centrifugal force related to rotating reference frames.

Is centrifugal force a "real" force?

The centrifugal and Coriolis forces are called "fictitious" because they do not appear in an inertial frame of reference. Unlike forces present in an inertial frame, fictitious forces do not express interactions between objects, like nuclear or electromagnetic forces. Instead, they are an expression of accelerations that make a frame non-inertial. In the non-inertial frame, these fictitious forces must be added to the real forces when using Newton's laws (and often are referred to as 'inertial forces' for that reason.) Because they do not originate from another object, there is no object to experience an associated reaction force: in this sense fictitious forces do not obey Newton's third law. However, with the addition of fictitious forces, Newton's laws do apply in non-inertial reference frames, such as planets, centrifuges, carousels, turning cars, and spinning buckets.

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.

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 equations describing these accelerations are (see fictitious force for a detailed discussion and derivation):

a r o t {\displaystyle \mathbf {a} _{\mathrm {rot} }\,} = a − 2 Ω × v r o t − Ω × ( Ω × r ) {\displaystyle =\mathbf {a} -2\mathbf {\Omega \times v_{\mathrm {rot} }} -\mathbf {\Omega \times (\Omega \times r)} \,}
= a + a c o r i o l i s + a c e n t r i f u g a l {\displaystyle =\mathbf {a+a_{\mathrm {coriolis} }+a_{\mathrm {centrifugal} }} \,} ,

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

a centrifugal = − Ω × ( Ω × r ) = ω 2 r ⊥ {\displaystyle \mathbf {a} _{\textrm {centrifugal}}=-\mathbf {\Omega \times (\Omega \times r)} =\omega ^{2}\mathbf {r} _{\perp }} ,

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

Fictitious forces

Main article: Fictitious force

An alternative way of dealing with a rotating frame of reference is to make Newton's laws of motion artificially valid by adding pseudo forces to be the cause of the above acceleration terms. In particular, the centrifugal acceleration is added to the motion of every object, and attributed to a centrifugal force, given by:

F c e n t r i f u g a l {\displaystyle \mathbf {F} _{\mathrm {centrifugal} }\,} = m a c e n t r i f u g a l {\displaystyle =m\mathbf {a} _{\mathrm {centrifugal} }\,}
= m ω 2 r ⊥ {\displaystyle =m\omega ^{2}\mathbf {r} _{\perp }\,}

where m {\displaystyle m\,} is the mass of the 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":

F c o r i o l i s = − 2 m Ω × v {\displaystyle \mathbf {F} _{\mathrm {coriolis} }=-2\,m\,{\boldsymbol {\Omega }}\times {\boldsymbol {v}}}

For example, a body that is stationary relative to the non-rotating frame will be rotating when viewed from the rotating frame. The centripetal force of − m ω 2 r ⊥ {\displaystyle -m\omega ^{2}\mathbf {r} _{\perp }} required to account for this apparent rotation is the sum of the centrifugal pseudo force m ω 2 r ⊥ {\displaystyle m\omega ^{2}\mathbf {r} _{\perp }} and the Coriolis force − 2 m Ω × v = − 2 m ω 2 r ⊥ {\displaystyle -2m{\boldsymbol {\Omega \times v}}=-2m\omega ^{2}\mathbf {r} _{\perp }} . Since this centripetal force includes contributions from only pseudo forces, it has no reactive counterpart.

Examples

Below several examples illustrate the role of centrifugal force and its relation to Coriolis force in rotating frameworks.

Rotating identical spheres

Figure 1: 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: 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 1 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 2. These two forces are provided by the string, putting the string under tension, also shown in Figure 2.

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, regardless of 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:

F c e n t r i p e t a l = − m ω 2 R   , {\displaystyle \mathbf {F} _{\mathrm {centripetal} }=-m\omega ^{2}R\ ,}

where ω is the angular rate of rotation, m is the mass of the ball, and R is the distance from the axis of rotation to 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:

F f i c t = − 2 m Ω × v B   {\displaystyle \mathbf {F} _{\mathrm {fict} }=-2m{\boldsymbol {\Omega }}\times \mathbf {v} _{B}\ }
= − 2 m ω ( ω R ) , {\displaystyle =-2m\omega \left(\omega R\right),}

where 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, R is the distance to the object from the center of rotation, and vB = ωR is the velocity of the object subject to the Coriolis force.

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.

Dropping ball

Figure 3: 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 3 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 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.

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.

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 3. 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

F f i c t = − m ω 2 R   , {\displaystyle \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.

Whirling table

Figure 4: 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 4 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.

Choice of frame of reference

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.

Intuition can go either way, and we can become perplexed when we switch viewpoints unconsciously.

The centrifuge supplies a similar example. 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.

Potential energy

Figure 5: 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 = − 1 2 m ω 2 r ⊥ 2 {\displaystyle 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 ) {\displaystyle h(r)\,} in a rotating bucket: requiring the potential energy per unit mass on the surface g h ( r ) − 1 2 ω 2 r 2 {\displaystyle gh(r)-{\frac {1}{2}}\omega ^{2}r^{2}\,} to be constant, we obtain the parabolic form h ( r ) = ω 2 2 g r 2 + C {\displaystyle h(r)={\frac {\omega ^{2}}{2g}}r^{2}+C} (where C {\displaystyle C} is a constant). See Figure 5.

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.

Applications

The operations of numerous common rotating mechanical systems are most easily conceptualized in terms of centrifugal force. For example: Template:Multicol

  • 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.

Template:Multicol-break

  • 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.

Template:Multicol-end 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

Notes and references

  1. "Centrifugal Force".
  2. "Centrifugal Force - Britannica online encyclopedia".
  3. Wolfram Scienceworld
  4. Stephen T. Thornton & Jerry B. Marion (2004). Classical Dynamics of Particles and Systems (5th Edition ed.). Belmont CA: Brook/Cole. p. Chapter 10. ISBN 0534408966. {{cite book}}: |edition= has extra text (help); line feed character in |author= at position 22 (help)
  5. (Marion & Thornton 1995, p. 386, 4th Edition) harv error: no target: CITEREFMarionThornton1995 (help)
  6. (Marion & Thornton 1995, p. 387 4th Edition) harv error: no target: CITEREFMarionThornton1995 (help)
  7. ^ John Robert Taylor (2004). Classical Mechanics. Sausalito CA: University Science Books. p. pp. 343-344. ISBN 189138922X. {{cite book}}: |page= has extra text (help)
  8. Max Born & Günther Leibfried (1962). Einstein's Theory of Relativity. New York: Courier Dover Publications. p. pp.76-78. ISBN 0486607690. {{cite book}}: |page= has extra text (help)
  9. NASA: Accelerated Frames of Reference: Inertial Forces
  10. Louis N. Hand & Janet D. Finch (1998). Analytical Mechanics. Cambridge UK: Cambridge University Press. p. pp. 266-267. ISBN 0521575729. {{cite book}}: |page= has extra text (help)
  11. John Robert Taylor (2004). Chapter 9, pp. 327 ff. ISBN 189138922X.
  12. The pull of hypergravity
  13. For more detail see Hall: Artificial gravity and the architecture of orbital habitats.
  14. Hall: Inhabiting artificial gravity
  15. See, for example, Pouly and Young.
  16. Vladimir Igorevich Arnolʹd (1989). Mathematical Methods of Classical Mechanics. Berlin: Springer. p. §27 pp. 129 ff. ISBN 0387968903.
  17. (Marion & Thornton 1995, pp. 386–387 4th Edition) harv error: no target: CITEREFMarionThornton1995 (help)
  18. 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.
  19. Dionysius Lardner (1877). Mechanics. Oxford University Press. p. p. 150. {{cite book}}: |page= has extra text (help)
  20. SI Rubinow (2002 (1975)). Introduction to mathematical biology. Courier/Dover Publications. p. pp. 235-244. ISBN 0486425320. {{cite book}}: |page= has extra text (help); Check date values in: |year= (help)CS1 maint: year (link)
  21. Jagannath Mazumdar (1999). An Introduction to Mathematical Physiology and Biology. Cambridge UK: Cambridge University Press. p. pp. 33 ff. ISBN 0521646758. {{cite book}}: |page= has extra text (help)

Further reading

External links

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