View source for Precession

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[[Image:Gyroscope precession.gif|thumb|right|Precession of a gyroscope]]
'''Precession''' refers to a change in the direction of the axis of a [[rotation|rotating]] object. In [[physics]], there are two types of precession, [[torque]]-free and torque-induced, the latter being discussed here in more detail. In certain contexts, "precession" may refer to the [[Precession (astronomy)|precession that the Earth experiences]], the effects of this type of precession on astronomical observation, or to the precession of orbital objects.

== Torque-free ==
Torque-free precession occurs when the axis of rotation differs slightly from an axis about which the object can rotate stably. [[Poinsot's construction]] is an elegant geometrical method for visualizing the torque-free motion of a rotating rigid body. For example, when a plate is thrown,  the plate may have some rotation around an axis that is not its [[Reflection symmetry|axis of symmetry]]. When an object is not perfectly [[solid]], internal [[vortices]] will tend to damp torque-free precession.

== Torque-induced ==
Torque-induced precession ('''gyroscopic precession''') is the phenomenon in which the [[axis of rotation|axis]] of a spinning object (e.g. a part of a [[gyroscope]]) "wobbles" when a [[torque]] is applied to it. The phenomenon is commonly seen in a [[spinning top|spinning toy top]], but all rotating objects can undergo precession. If the [[speed]] of the rotation and the [[Magnitude (mathematics)|magnitude]] of the torque are constant the axis will describe a [[Cone (geometry)|cone]], its movement at any instant being at [[right angle]]s to the [[Direction (geometry, geography)|direction]] of the torque. In the case of a toy top, if the axis is not perfectly vertical the torque is applied by the [[force (physics)|force]] of [[gravity]] tending to tip it over. 

[[Image:Gyroscopic precession 256x256.png|frame|right|The response of a rotating system to an applied torque. When the device swivels, and some roll is added, the wheel tends to pitch.]]

The device depicted on the right here is [[gimbal]] mounted. From inside to outside there are three axes of rotation: the hub of the wheel, the gimbal axis and the vertical pivot.

To distinguish between the two horizontal axes, rotation around the wheel hub will be called 'rolling', and rotation around the gimbal axis will be called 'pitching.'  Rotation around the pivot axis is called 'spinning'.

First, imagine that the device is spinning around the pivot axis. Then some rotation around the wheelhub is added. Imagine the gimbal axis to be locked, so that the wheel cannot pitch. The gimbal axis has sensors, that measure whether there is a [[torque]] around the gimbal axis. 

In the picture, a section of the wheel has been named 'dm1'. When the rolling starts, section dm1 is at the [[perimeter]] of the spinning motion. Section dm1 has a lot of [[velocity]] and as it is forced closer to the center of rotation, it tends to move in the direction of the top-left arrow in the diagram (shown at 45<sup>o</sup> in the direction of rolling). Section dm2 of the wheel starts out at the center of rotation, and thus initially has zero velocity before the wheel is rolled. A force would be required to increase section dm2's velocity to the velocity at the perimeter of the pivot axis' plane. If that force is not provided then section dm2's [[inertia]] will make it move in the direction of the top-right arrow.  Note that both arrows point in the same direction.

The same reasoning applies for the bottom half of the wheel, but there the arrows point in the opposite direction to that of the top arrows. Combined over the entire wheel, there is a torque around the gimbal axis when some rolling is added to rotation around a vertical axis. 

It is important to note that the torque around the gimbal axis arises without any delay; the response is instantaneous. 

In the discussion above, the setup was kept unchanging by preventing rotation around the gimbal axis. In the case of a spinning top, when the spinning top is tilting, gravity exerts a torque. Instead of rolling over, the spinning top pitches. The pitching motion reorients the spinning top with respect to the torque that is being exerted. The result is that the torque exerted by gravity elicits gyroscopic precession rather than causing the spinning top to fall to its side. 

Precession or gyroscopic considerations have an effect on [[bicycle]] performance at high speed. Precession is also the mechanism behind [[gyrocompass]]es. 

Gyroscopic precession also plays a large role in the [[Aircraft flight control systems|flight controls]] on [[helicopter]]s.  Since the driving force behind helicopters is the [[Helicopter rotor|rotor disk]] (which rotates), gyroscopic precession comes into play.  If the rotor disk is to be tilted forward (to gain forward velocity), its rotation requires that the downward [[net force]] on the blade be applied roughly 90 degrees (depending on blade configuration) before, or when the blade is to one side of the pilot and rotating forward.  

To ensure the pilot's inputs are correct, the aircraft has corrective linkages which vary the blade pitch in advance of the blade's position relative to the [[swashplate]].  Although the swashplate moves in the intuitively correct direction, the blade pitch links are arranged to transmit the pitch in advance of the blade's position.

== Physics ==
=== Classical (Newtonian) ===
[[Image:PrecessionOfATop.svg|thumb|right|300px|The [[torque]] caused by the two opposing forces '''F'''<sub>g</sub> and -'''F'''<sub>g</sub> causes a change in the [[angular momentum]] '''L''' in the direction of that torque. This causes the top to precess.]]

Precession is the result of the [[angular velocity]] of rotation and the angular velocity produced by the torque. It is an angular velocity about a line which makes an angle with the permanent rotation axis, and this angle lies in a [[Angular momentum|plane]] at right angles to the plane of the couple producing the torque. The permanent axis must turn towards this line, since the body cannot continue to rotate about any line which is not a principal axis of maximum [[moment of inertia]]; that is, the permanent axis turns in a direction at right angles to that in which the torque might be expected to turn it. If the rotating body is [[Symmetry|symmetrical]] and its motion unconstrained, and if the torque on the spin axis is at right angles to that axis, the axis of precession will be perpendicular to both the spin axis and torque axis. 

Under these circumstances the angular velocity of precession is given by:

:<math>
\boldsymbol\omega_p = \frac{Q}{I_s\boldsymbol\omega_s}
</math>

In which ''I<sub>s</sub>'' is the [[moment of inertia]], <math>\boldsymbol\omega_s</math> is the angular velocity of spin about the spin axis, and ''Q'' is the torque. Using <math>\boldsymbol\omega</math> = <math>\frac{2\pi}{T}</math>, we find that the [[Frequency|period]] of precession is given by:

:<math>
T_p = \frac{4\pi^2I_s}{QT_s}
</math>

In which ''I<sub>s</sub>'' is the [[moment of inertia]], ''T<sub>s</sub>'' is the period of spin about the spin axis, and ''Q'' is the [[torque]]. In general the problem is more complicated than this, however.

'''An informal explanation of Precession:'''
In a classic beginning physics demonstration, the instructor stands on a swiveling platform and holds a spinning [[bicycle wheel]] at arm's length.  The wheel is [[Vertical direction|vertical]] and the instructor is standing still.  The instructor then tilts the wheel toward [[Horizontal plane|horizontal]].  This causes the instructor to start spinning slowly on the platform<!-- Which way?? -->.  Bringing the wheel back to vertical and tilting it the other way makes the instructor spin the other way.  Why?

Imagine the wheel as a collection of small particles.  Particles want to move in a straight line.  In order for them to move in a circle there must be a force accelerating the particles toward the center of the circle ([[acceleration]] is a change in speed or direction or both &mdash; in this case just direction).  This force is ultimately provided by bonds between the atoms in the wheel and [[spoke]]s.

What happens when the instructor turns the spinning wheel from vertical to horizontal? Consider a particle somewhere on the wheel. If the wheel weren't being tilted, it would be accelerated around the circle as always.  But since the wheel is tilting, it now has to follow a new path.  A change in path is an acceleration, which in turn requires force (from the instructor's hands, transmitted through the spokes to the rim).  Now consider the particle opposite the first particle on the wheel.   It also has to change path, but in the opposite direction.  Since the forces on opposite sides are in opposite directions, the result is  torque.  Each pair of opposite particles on the wheel contributes to the torque that causes the instructor to turn on the platform.  <!-- To really nail this down, we'd have to explain why they don't cancel each other out, that is why the net torque is non-zero -->

Tilting the wheel the other direction produces torque in the opposite direction, slowing the instructor's spin and eventually reversing it.

=== Relativistic ===
The special and general theories of [[Theory of relativity|relativity]] give three types of corrections to the Newtonian precession, of a gyroscope near a large mass such as the earth, described above. They are:
* [[Thomas precession]] a special relativistic correction accounting for the observer being in a rotating non-inertial frame.
* [[de Sitter precession]] a general relativistic correction accounting for the schwarzschild metric of curved space near a large non-rotating mass.
* [[Lense-Thirring precession]] a general relativistic correction accounting for the frame dragging by the Kerr metric of curved space near a large rotating mass.

== Of the Earth's axis ==
{{Main|Precession (astronomy)}}
[[Image:Earth precession.svg|thumb|right|Precessional movement.]]
The Earth goes through one complete '''precession''' cycle in a period of approximately 25,800 years, during which the positions of [[star]]s as measured in the [[equatorial coordinate system]] will slowly change; the change is actually due to the change of the coordinates. Over this cycle the Earth's north axial pole moves from where it is now, within 1° of [[Polaris]], in a circle around the [[ecliptic pole]], with an angular radius of about 23.5 degrees (or approximately 23 degrees 27 arcminutes <ref> {{cite web | last=Cook |first=David R. | year=1999 | url= http://www.newton.dep.anl.gov/askasci/env99/env154.htm |title=Tilt of Earth's Axis |work=Environmental Earth Science Archive, Ask A Scientist |publisher=[[United States Department of Energy]] |accessdate=2006-05-24}} </ref>). The shift is 1 degree in 180 years, where the angle is taken from the observer, not from the center of the circle.

Discovery of the precession of the equinoxes is generally attributed to the ancient Greek astronomer [[Hipparchus]] (ca. 150 B.C.), though the difference between the [[sidereal year|sidereal]] and [[tropical year|tropical]] years was known to [[Aristarchus of Samos]] much [http://www.dioi.org/cot.htm#fwkp earlier] (ca. 280 B.C.). It was later explained by [[classical mechanics|Newtonian physics]]. The Earth has a nonspherical shape, being [[Oblate|oblate spheroid]], bulging outward at the equator. The gravitational [[tidal force]]s of the [[Moon]] and [[Sun]] apply torque as they attempt to pull the [[equatorial bulge]] into the plane of the [[ecliptic]]. The portion of the precession due to the combined action of the Sun and the Moon is called '''lunisolar precession'''.

[[Orbit|Revolution]] of a planet in its [[orbit]] around the [[Sun]] is also a form of rotary motion.  (In this case, the combined system of Earth and Sun is rotating.)  So the axis of a planet's orbital plane will also precess over time. 

The major axis of each planet's elliptical orbit also precesses within its orbital plane, partly in response to perturbations in the form of the changing gravitational forces exerted by other planets. This is called '''perihelion precession''' or '''apsidal precession''' (see [[apsis]]). Discrepancies between the observed perihelion precession rate of the planet [[Mercury (planet)|Mercury]] and that predicted by [[classical mechanics]] were prominent among the forms of experimental evidence leading to the acceptance of [[Albert Einstein|Einstein]]'s [[Theory of Relativity]], which predicted the anomalies accurately.<ref>[[Max Born]] ([[1924]]), ''Einstein's Theory of Relativity'' (The 1962 Dover edition, page 348 lists a table documenting the observed and calculated values for the precession of the perihelion of Mercury, Venus, and Earth.)</ref><ref>[http://www.dailygalaxy.com/my_weblog/2008/03/18-billion-suns.html An even larger value for a precession has been found, for a black hole in orbit around a much more massive black hole, amounting to 39 degrees each orbit.]</ref>

These periodic changes of Earth's orbital parameters, combined with the precession of the equinoxes and of the [[inclination]] of the Earth's axis on its orbit, are an important part of the [[Milankovitch cycles|astronomical theory]] of [[ice age]]s.  For precession of the lunar orbit see [[lunar precession]].

A phenomenon analogous to apsidal precession is '''nodal precession''' (see [[orbital node]]), which affects the orientation of the orbital plane.  

{{commons|Category:Precession|Precession}}

== See also ==

*[[De Sitter precession]]
*[[Larmor precession]]
*[[Lense-Thirring precession]]
*[[Nutation]]
*[[Polar motion]]
*[[Precession (astronomy)]]
*[[Precession (mechanical)]]
*[[Thomas precession]]
*[[Euler angles]]

== References ==
{{reflist|2}}

* "Moon and Spica", StarDate July 14, 2005, University of Texas McDonald Observatory, [http://stardate.org/radio/program.php?f=detail&id=20050714]

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