User:FyzixFighter/Centrifugal force

This page is about the inertial force appearing in non-inertial reference frames. For the outward reaction force to a centripetal force, see reactive centrifugal force.

Centrifugal force (from Latin centrum "center" and fugere "to flee") can generally be any force directed outward relative to some origin. More particularly, in classical mechanics, centrifugal force is an outward force which arises when describing the motion of objects in a rotating reference frame. Because a rotating frame is an example of a non-inertial reference frame, Newton's laws of motion cannot be used to describe the dynamics within the frame. However, a rotating frame can be treated as if it were an inertial frame where Newton's laws are valid if so-called fictitious forces (also known as inertial or pseudo- forces) are included in the sum of external forces on an object.

Analysis of motion within rotating frames can be greatly simplified by the use of the fictitious forces. By starting with an inertial frame, where Newton's laws of motion hold, and seeing how the time derivatives of a position vector change when transforming to a rotating reference frame, the various fictitious forces and their forms can be identified. Rotating frames and fictitious forces can often reduce the description of motion in two dimensions to a simpler description in one dimensions (corresponding to a co-rotating frame). In this approach, circular motion in an inertial frame, which only requires the presence of a centripetal force, becomes the balance between the real centripetal force and the frame-determined centrifugal force in a rotating frame where the object appears stationary. Also in this approach, if a rotating frame is chosen so that just the angular position of an object is held fixed, more complicated radial motion, like that of elliptical and open orbits, appears when the centripetal and centrifugal forces do not balance. The general approach however is not limited to these co-rotating frames, but can be equally applied to objects at motion in any rotating frame. All objects in any rotating frame will appear to experience the outward centrifugal force.

The centrifugal force is what is usually thought of as the cause for apparent outward movement, like that of passengers in a vehicle turning a corner, of the weights in a centrifugal governor, and of particles in a centrifuge. From the standpoint of an observer in an inertial frame, the effects can be explained as results of inertia without invoking the centrifugal force. Centrifugal force should not be confused with centripetal force or the reactive centrifugal force, both of which are real forces independent of the frame of the observer.

History

Main article: History of centrifugal and centripetal forces

The conception of centrifugal force has evolved since the time of Huygens, Newton, Leibniz, and Hooke who expressed early conceptions of it. The modern conception as a fictitious force or pseudo force due to a rotating reference frame as described above evolved in the eighteenth and nineteenth centuries.

• needs better summary of history article - more than two sentences
• Something about Mach?

In classical Newtonian mechanics

Although Newton's laws of motion hold exclusively in inertial frames, often times is far more convenient and more advantageous to describe the motion of objects within a rotating 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). By treating the extra acceleration terms due to the rotation of the frame as if they were forces, subtracting them from the physical forces, it's possible to treat the second time derivative of position (relative to the rotating frame) as absolute acceleration. Thus the analysis using Newton's laws of motion can proceed as if the reference frame was inertial, provided the fictitious force terms are included in the sum of external forces. For 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.

A disadvantage of a rotating reference frame is that it can be more difficult to apply special relativity (for example, from the perspective of the Earth the stars seem to traverse many light-years each day). It is possible to do so if a metric tensor is introduced, but the speed of light may not be constant and clocks within the frame are not synchronized.

There are three general scenarios in which this concept of a fictitious centrifugal force arises when describing motion:

1. When the motion is described relative to a rotating reference frame about a fixed axis at the origin of the coordinate system. For observations made in the rotating frame, all objects appear to be under the influence of a radially outward force that is proportional to the distance from the axis of rotation and to the square of the rate of rotation (angular velocity) of the frame.
2. When the motion is described using an accelerated local reference frame attached to a moving body, for example, the frame of passengers in a car as it rounds a corner. In this case, rotation is again involved, this time about the center of curvature of the path of the moving body. The first context can be seen as a special scenario within this second context in which the origin of the coordinate system and the axis of rotation are always coincident. In both, the centrifugal force is zero when the rate of rotation of the reference frame is zero, independent of the motions of objects in the frame.
3. The third context is the most general, and subsumes the first two, as well as stationary curved coordinates (e.g., polar coordinates). The centrifugal force appears when the terms for the radial component of the equation of motion are rearranged to resemble Newton's second law for one-dimensional motion. Therefore, the centrifugal force is simply the sign-reversal of the centripetal acceleration for motion along curves where the radial distance is fixed and is related to the Christoffel symbol term related to that curvature. While no rotation is necessary in this derivation, reapplying Newtonian definitions of force and acceleration to the rearranged equation necessarily implies observing the motion from a co-rotating frame of reference.

In each of these scenarios, the centrifugal force is an inertial force used for convenience and implied by a specific, non-inertial reference frame.

If objects are seen as moving within a rotating frame, this movement results in another fictitious force, the Coriolis force; and if the rate of rotation of the frame is changing, a third fictitious force, the Euler force is experienced. Together, these three fictitious forces allow for the creation of correct equations of motion in a rotating reference frame.

Derivation

In a rotating frame of reference the time derivatives of any position vector r, such as the velocity and acceleration vectors, will differ from the time derivatives in an inertial frame according to the frame's rotation. The first time derivative [dr/dt] evaluated from a reference frame with a coincident origin at $r=0$ but rotating with the absolute angular velocity Ω is:

${\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}=\left[{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\right]+{\boldsymbol {\Omega }}\times {\boldsymbol {r}}\ ,$

where $\times$ denotes the vector cross product and square brackets […] denote evaluation in the rotating frame of reference. In other words, the apparent velocity in the rotating frame is altered by the amount of the apparent rotation ${\boldsymbol {\Omega }}\times {\boldsymbol {r}}$ at each point, which is perpendicular to both the vector from the origin r and the axis of rotation Ω and directly proportional in magnitude to each of them. The vector Ω has magnitude Ω equal to the rate of rotation and is directed along the axis of rotation according to the right-hand rule.

Newton's law of motion for a particle of mass m written in vector form is

${\boldsymbol {F}}=m{\boldsymbol {a}}\ ,$

where F is the vector sum of the physical forces applied to the particle and a is the absolute acceleration of the particle, given by:

${\boldsymbol {a}}={\frac {\operatorname {d} ^{2}{\boldsymbol {r}}}{\operatorname {d} t^{2}}}\ ,$

where r is the position vector of the particle.

By twice applying the transformation above from the inertial to the rotating frame, the absolute acceleration of the particle can be written as:

${\begin{aligned}{\boldsymbol {a}}&={\frac {\operatorname {d} ^{2}{\boldsymbol {r}}}{\operatorname {d} t^{2}}}={\frac {\operatorname {d} }{\operatorname {d} t}}{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}={\frac {\operatorname {d} }{\operatorname {d} t}}\left(\left[{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\right]+{\boldsymbol {\Omega }}\times {\boldsymbol {r}}\ \right)\\&=\left[{\frac {\operatorname {d} ^{2}{\boldsymbol {r}}}{\operatorname {d} t^{2}}}\right]+{\frac {\operatorname {d} {\boldsymbol {\Omega }}}{\operatorname {d} t}}\times {\boldsymbol {r}}+2{\boldsymbol {\Omega }}\times \left[{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\right]+{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times {\boldsymbol {r}})\ .\end{aligned}}$

The apparent acceleration in the rotating frame is [d²r/dt²]. An observer unaware of the rotation would expect this to be zero in the absence of outside forces. However Newton's laws of motion apply only in the inertial frame and describe motion in terms of the absolute acceleration d²r/dt². Therefore the observer perceives the extra terms as contributions due to fictitious forces. These terms in the apparent acceleration are independent of mass; so it appears that each of these fictitious forces, like gravity, pulls on an object in proportion to its mass. When these forces are added, the equation of motion has the form:

${\boldsymbol {F}}-m{\frac {\operatorname {d} {\boldsymbol {\Omega }}}{\operatorname {d} t}}\times {\boldsymbol {r}}-2m{\boldsymbol {\Omega }}\times \left[{\frac {\operatorname {d} \mathbf {r} }{\operatorname {d} t}}\right]-m{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times {\boldsymbol {r}})$ $=m\left[{\frac {\operatorname {d} ^{2}{\boldsymbol {r}}}{\operatorname {d} t^{2}}}\right]\ .$

From the viewpoint of the rotating frame, the additional force terms are experienced just like the real external forces and contribute to the apparent acceleration. The additional terms on the force side of the equation can be recognized as, reading from left to right, the Euler force, the Coriolis force, and the centrifugal force, respectively. Unlike the other two fictitious forces, the centrifugal force always points directly away from the axis of rotation of the rotating reference frame, with magnitude mΩ²r, and unlike the Coriolis force in particular, the centrifugal force is independent of the motion of the particle in the rotating frame. As expected, for a non-rotating inertial frame of reference $({\boldsymbol {\Omega }}=0)$ the centrifugal force and all other fictitious forces disappear.

Examples

Stationary objects

Vector cross product used to determine the Coriolis force.

Describing the force on objects at rest relative to a frame of reference rotating at a constant rate only requires the additional consideration of a centrifugal force. However, the fictitious forces on objects moving in the rotating frame also includes the Coriolis force. In the figure, 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).

A simple example of body observed to be moving in a rotating frame is that of a body that is stationary relative to a non-rotating inertial. Viewed from the rotating frame, the body follows a circular path and therefore, by application of Newton's laws to what looks like circular motion in the rotating frame at a radius r, requires an inward force of −m Ω² r, where Ω is angular rate of rotation of the frame. This centripetal force in the rotating frame is provided as a net fictitious force that is the sum of the radially outward centrifugal force m Ω² r and the Coriolis force −2m Ω × v_rot. To evaluate the Coriolis force, we need the velocity as seen in the rotating frame, v_rot, which is given by −Ω × r. This results in a Coriolis force with a value −2m Ω² r, the negative sign indicating that it is radially inward. 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.

Planetary motion

In the analysis of orbital motion, the symmetry of a central force lends itself to a description in polar coordinates. The dynamics of a mass, m, expressed using Newton's second law of motion (F = ma), becomes in polar coordinates:

${\boldsymbol {F}}=m(({\ddot {r}}-r{\dot {\theta }}^{2}){\boldsymbol {\hat {r}}}+(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}){\boldsymbol {\hat {\theta }}})$

where ${\boldsymbol {F}}$ is the force acting on the object and can be expressed, because it is a central force, as $F(r){\boldsymbol {\hat {r}}}$. The sign of F(r) indicates whether the central force is inward (negative) or outward (positive) - gravity, an inward central force, would correspond to a negative-valued F(r).

The components of F = ma along the radial direction therefore reduce to

$F(r)=m({\ddot {r}}-r{\dot {\theta }}^{2})$

in which the term proportional to the square of the rate of rotation appears on the acceleration side is sometimes called a "centripetal acceleration", that is, a negative acceleration term in the ${\boldsymbol {\hat {r}}}$ direction. In the special case of a planet in circular orbit around its star, for example, where ${\ddot {r}}$ is zero, the centripetal acceleration alone is the entire acceleration of the planet.

It is sometimes convenient to work in a co-rotating frame, that is, one rotating with the object so that the angular rate of the frame, $\Omega$, equals the ${\dot {\theta }}$ of the object in the inertial frame. In such a frame, the observed ${\dot {\theta }}$ is zero and ${\ddot {r}}$ alone is treated as the acceleration. Rearranging the terms in the equation for the radial components of F=ma to isolate the radial acceleration in the co-rotating frame yields

$F(r)+mr{\dot {\theta }}^{2}=m{\ddot {r}}$.

Expressing the quantities in terms of rotating frame coordinates and the rotation rate of the frame, replacing ${\dot {\theta }}$ with $\Omega$, shows that the $mr{\dot {\theta }}^{2}$ term which was originally on the acceleration side of the equation reappears on the force side of the equation (with opposite signs, of course) as the centrifugal force mΩ²r in the radial equation for the co-rotating frame.

Because of the absence of a net force in the azimuthal direction, conservation of angular momentum allows the radial component of this equation to be expressed solely with respect to the radial coordinate, r, and the angular momentum $L=m{\dot {\theta }}r^{2}$, yielding the radial equation (a "fictitious one-dimensional problem" with only an r dimension):

$F(r)+{\frac {L^{2}}{mr^{3}}}=m{\ddot {r}}$

with the centrifugal force corresponding to the $L^{2}/mr^{3}$ term. The equations of motion for r that result from this equation for the rotating two-dimensional frame are the same that would arise from a particle in a fictitious one-dimensional scenario under the influence of an effective force that is the sum of F(r) and the centrifugal force.

If F(r) represents gravity, it is a negative term proportional to 1/r². The solutions to the radial equation include both bound solutions, corresponding to circular and elliptical orbits, and unbound solutions, corresponding to parabolic and hyperbolic trajectories. When the angular velocity of this co-rotating frame is not constant, that is, for non-circular orbits, other fictitious forces – the Coriolis force and the Euler force – will arise, but can be ignored since they will cancel each other, yielding a net zero acceleration transverse to the moving radial vector, as required by the starting assumption that the ${\hat {r}}$ vector co-rotates with the planet. In the special case of circular orbits, in order for the radial distance to remain constant the outward centrifugal force must cancel the inward force of gravity; for other orbit shapes, these forces will not cancel, so r will not be constant.

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.

Absolute rotation

Main article: Absolute rotation

Three experiments were suggested by Newton to answer the question of whether the absolute rotation of a local frame, that is the frame in which an object is stationary, can be detected. The first experiment relies on the effect of centrifugal force upon the shape of the surface of water rotating in a bucket. The second looks at the effect of centrifugal force upon the tension in a string joining two spheres rotating about their center of mass. The tension in the string will be proportional to the centrifugal force on each sphere as it rotates around the common center of mass. The third uses the effect of centrifugal force on the oblateness of a sphere of freely flowing material. The oblate spheroid shape reflects, following Clairaut's theorem, the balance between containment by gravitational attraction and dispersal by centrifugal force. In these experiments, the effects attributed to centrifugal force are only observed in the local frame if the objects are undergoing absolute rotation relative to an inertial frame. By contrast, in an inertial frame, the observed effects arise as a consequence of the inertia of the objects without the need to introduce a centrifugal force.

In Lagrangian mechanics

Lagrangian mechanics formulates mechanics in terms of generalized coordinates $\{q_{k}\}$, which can be as simple as the usual polar coordinates $(r,\ \theta )$ or a much more extensive list of variables. Within this formulation the motion is described in terms of generalized forces, using in place of Newton's laws the Euler-Lagrange equations. Among the generalized forces, those involving the square of the time derivatives $\{(\mathrm {d} q_{k}/\mathrm {d} t)^{2}\}$ are sometimes called centrifugal forces.

For the particular case of single-body motion found using the polar coordinates $({\dot {r}},\ {\dot {\theta }})$ as the generalized coordinates in a central force, the Euler–Lagrange equations are the same equations found using Newton's second law in a co-rotating frame. For example, the radial equation is:

$\mu {\ddot {r}}=\mu r{\dot {\theta }}^{2}-{\frac {\mathrm {d} U}{\mathrm {d} r}}$

where $U(r)$ is the central force potential and μ is the mass of the object. The left side is a "generalized force" and the first term on the right is the "generalized centrifugal force". However, the left side is not comparable to a Newtonian force, as it does not contain the complete acceleration, and likewise, therefore, the terms on the right-hand side are "generalized forces" and cannot be interpreted as Newtonian forces. In the special case of motion in a central potential described with polar coordinates, the Lagrangian centrifugal force is the same as the fictitious centrifugal force derived in a co-rotating frame. However, the Lagrangian use of "centrifugal force" in other, more general cases has only a limited connection to the Newtonian definition.