Editing Precession (section)

==Torque-induced==
Torque-induced precession ('''gyroscopic precession''') is the phenomenon in which the [[axis of rotation|axis]] of a spinning object (e.g., a [[gyroscope]]) describes a [[Cone (geometry)|cone]] in space when an external [[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 external torque are constant, the spin axis will move at [[right angle]]s to the [[Direction (geometry, geography)|direction]] that would intuitively result from the external torque. In the case of a toy top, its weight is acting downwards from its [[center of mass]] and the [[normal force]] (reaction) of the ground is pushing up on it at the point of contact with the support. These two opposite forces produce a torque which causes the top to precess.

[[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 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 ''spinning'', and rotation around the gimbal axis will be called ''pitching''.  Rotation around the vertical pivot axis is called ''rotation''.

First, imagine that the entire device is rotating around the (vertical) pivot axis. Then, spinning of the wheel (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 {{math|''dm''<sub>1</sub>}}. At the depicted moment in time, section {{math|''dm''<sub>1</sub>}} is at the [[perimeter]] of the rotating motion around the (vertical) pivot axis. Section {{math|''dm''<sub>1</sub>}}, therefore, has a lot of angular rotating [[velocity]] with respect to the rotation around the pivot axis, and as {{math|''dm''<sub>1</sub>}} is forced closer to the pivot axis of the rotation (by the wheel spinning further), because of the [[Coriolis effect]], with respect to the vertical pivot axis, {{math|''dm''<sub>1</sub>}} tends to move in the direction of the top-left arrow in the diagram (shown at 45°) in the direction of rotation around the pivot axis.<ref name="Teodorescu">{{cite book | last = Teodorescu | first = Petre P | title = Mechanical Systems, Classical Models: Volume II: Mechanics of Discrete and Continuous Systems | publisher = Springer Science & Business Media | date = 2002 | page = 420 | url = https://books.google.com/books?id=aXCBlHOtO3kC&pg=PA396 | isbn = 978-1-4020-8988-6}}</ref>  Section {{math|''dm''<sub>2</sub>}} of the wheel is moving away from the pivot axis, and so a force (again, a Coriolis force) acts in the same direction as in the case of {{math|''dm''<sub>1</sub>}}. 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 spinning 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 pitching around the gimbal axis. In the case of a spinning toy top, when the spinning top starts tilting, gravity exerts a torque. However, instead of rolling over, the spinning top just pitches a little. This pitching motion reorients the spinning top with respect to the torque that is being exerted. The result is that the torque exerted by gravity – via the pitching motion – elicits gyroscopic precession (which in turn yields a counter torque against the gravity torque) 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.

===Classical (Newtonian)===

[[File:PrecessionOfATop.svg|thumb|right|256px|The [[torque]] caused by the normal force – {{math|'''F'''<sub>g</sub>}} and the weight of the top causes a change in the [[angular momentum]] {{math|'''L'''}} in the direction of that torque. This causes the top to precess.]]

Precession is the change of [[angular velocity]] and [[angular momentum]] produced by a torque. The general equation that relates the torque to the rate of change of angular momentum is:
<math display="block">\boldsymbol{\tau} = \frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t}</math>
where <math>\boldsymbol{\tau}</math> and <math>\mathbf{L}</math> are the torque and angular momentum vectors respectively.

Due to the way the torque vectors are defined, it is a vector that is perpendicular to the plane of the forces that create it. Thus it may be seen that the angular momentum vector will change perpendicular to those forces. Depending on how the forces are created, they will often rotate with the angular momentum vector, and then circular precession is created.

Under these circumstances the angular velocity of precession is given by: <ref>{{cite book |last1=Moebs |first1=William |last2=Ling |first2=Samuel J. |last3=Sanny |first3=Jeff |title=11.4 Precession of a Gyroscope - University Physics Volume 1 {{!}} OpenStax |date=Sep 19, 2016 |location=Houston, Texas |url=https://openstax.org/books/university-physics-volume-1/pages/11-4-precession-of-a-gyroscope |access-date=23 October 2020 |language=en}}</ref>

:<math>\boldsymbol\omega_\mathrm{p} = \frac{\ mgr}{I_\mathrm{s}\boldsymbol\omega_\mathrm{s}} = \frac{ \tau}{I_\mathrm{s}\boldsymbol\omega_\mathrm{s}\sin(\theta)}</math>

where {{math|''I''<sub>s</sub>}} is the [[moment of inertia]], {{math|'''''ω'''''<sub>s</sub>}} is the angular velocity of spin about the spin axis, {{mvar|m}} is the mass, {{math|''g''}} is the acceleration due to gravity, {{mvar|θ}} is the angle between the spin axis and the axis of precession and {{math|''r''}} is the distance between the center of mass and the pivot.  The torque vector originates at the center of mass. Using {{math|1='''''ω''''' = {{sfrac|2π|''T''}}}}, we find that the [[Frequency|period]] of precession is given by:<ref>{{cite book |last1=Moebs |first1=William |last2=Ling |first2=Samuel J. |last3=Sanny |first3=Jeff |title=11.4 Precession of a Gyroscope - University Physics Volume 1 {{!}} OpenStax |date=Sep 19, 2016 |location=Houston, Texas |url=https://openstax.org/books/university-physics-volume-1/pages/11-4-precession-of-a-gyroscope |access-date=23 October 2020 |language=en}}</ref>
<math display="block">T_\mathrm{p} = \frac{4\pi^2 I_\mathrm{s}}{\ mgrT_\mathrm{s}} = \frac{4\pi^2 I_\mathrm{s}\sin(\theta)}{\ \tau T_\mathrm{s}}</math>

Where {{math|''I''<sub>s</sub>}} is the [[moment of inertia]], {{math|''T''<sub>s</sub>}} is the period of spin about the spin axis, and {{mvar|'''τ'''}} is the [[torque]]<!-- Torque is not introduced -->. In general, the problem is more complicated than this, however.

===Relativistic (Einsteinian) ===

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 Earth, described above. They are:
* [[Thomas precession]], a special-relativistic correction accounting for an object (such as a gyroscope) being accelerated along a curved path.
* [[Geodetic effect|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.

The [[Schwarzschild geodesics]] (sometimes Schwarzschild precession) is used in the prediction of the [[anomalous perihelion precession]] of the planets, most notably for the accurate prediction of the [[Precession#Apsidal precession|apsidal precession]] of Mercury.