Editing Gyroscope (section)

== Gyroscopic principles ==
All spinning objects have gyroscopic properties. The main properties that an object can experience in any gyroscopic motion are [[Axial parallelism|rigidity in space]] and [[precession]].

=== Rigidity in space ===
{{main|Axial parallelism}}
Rigidity in space describes the principle that a gyroscope remains in the fixed position on the plane in which it is spinning, unaffected by the Earth's rotation. For example, a bike wheel. Early forms of gyroscope (not then known by the name) were used to demonstrate the principle.<ref name="The Encyclopaedia Britannica: A Dictionary of Arts, Sciences and General Literature 1890 p. 351">{{cite book | title=The Encyclopaedia Britannica: A Dictionary of Arts, Sciences and General Literature | publisher=R.S. Peale | issue=v. 11 | year=1890 | url=https://books.google.com/books?id=wqwMAAAAYAAJ&pg=PA351 | access-date=2022-12-02 | page=351|quote=Under the title of precession instruments, various pieces of apparatus, involving the gyroscope principle, have been in use for a number of years for illustrating the precession of the equinoxes, and the parallelism of the earth's axis as it revolves round the sun.}}</ref>

===Precession===
{{main|Axial precession}}
A simple case of precession, also known as steady precession, can be described by the following relation to Moment:

:<math>\sum M_x = -I{\phi'}^2 \sin\theta \cos\theta +I_z\phi' \sin\theta(\phi' \cos\theta + \psi' ) </math>

where  <math>\phi'</math> represents precession,  <math>\psi'</math> is represented by spin, <math>\theta</math> is the nutation angle, and <math>I</math> represents inertia along its respective axis. This relation is only valid with the Moment along the Y and Z axes are equal to 0.

The equation can be further reduced noting that the angular velocity along the z-axis is equal to the sum of the Precession and the Spin: <math>\omega_z = \phi' \cos \theta + \psi'</math>, Where <math>\omega_z </math> represents the angular velocity along the z axis.

:<math>\sum M_x =  -I{\psi'}^2 \sin \theta \cos \theta + I_z \psi' (\sin\theta)\omega_z
</math>

or

:<math>\sum M_x = \psi' \sin \theta (I_z\omega_z-I\psi' \cos \theta)</math><ref>{{Cite book|last=Hibbeler|first=R.C|title=Engineering Mechanics: Dynamics Fourteenth Edition|publisher=[[Pearson Prentice Hall]]|year=2016|location=Hoboken, New Jersey|pages=627–629}}</ref>{{full citation needed|reason=ISBN needed|date=July 2022}}

Gyroscopic [[precession]] is torque induced. It is the rate of change of the angular momentum that is produced by the applied torque. Precession produces counterintuitive dynamic results such as a [[Top (toy)|spinning top]] not falling over. Precession is used in aerospace applications for sensing changes of attitude and direction.