Editing Angular momentum (section)

=== Orbital angular momentum in two dimensions ===
[[File:Ang mom 2d.png|thumb|[[Velocity]] of the [[particle]] ''m'' with respect to the origin ''O'' can be resolved into components parallel to (''v''<sub>∥</sub>) and perpendicular to (''v''<sub>⊥</sub>) the radius vector ''r''. The '''angular momentum''' of ''m'' is proportional to the [[perpendicular component]] ''v''<sub>⊥</sub> of the velocity, or equivalently, to the perpendicular distance ''r''<sub>⊥</sub> from the origin.]]

Angular momentum is a [[Euclidean vector|vector]] quantity (more precisely, a [[pseudovector]]) that represents the product of a body's [[moment of inertia|rotational inertia]] and [[angular velocity|rotational velocity]] (in radians/sec) about a particular axis. However, if the particle's trajectory lies in a single [[plane (geometry)|plane]], it is sufficient to discard the vector nature of angular momentum, and treat it as a [[scalar (mathematics)|scalar]] (more precisely, a [[pseudoscalar]]).<ref name="Wilson">
{{cite journal
 |last1 = Wilson
 |first1 = E. B.
 |title = Linear Momentum, Kinetic Energy and Angular Momentum
 |journal=The American Mathematical Monthly
 |volume=XXII
 |publisher = Ginn and Co., Boston, in cooperation with University of Chicago, et al.
 |date=1915
 |url=https://books.google.com/books?id=nsI0AAAAIAAJ |page= 190
}}</ref> Angular momentum can be considered a rotational analog of linear momentum. Thus, where linear momentum {{mvar|p}} is proportional to [[mass]] {{mvar|m}} and [[speed|linear speed]] {{nowrap|{{mvar|v}},}}
<math display="block" qid=Q41273>p = mv,</math>
angular momentum {{mvar|L}} is proportional to [[moment of inertia]] {{mvar|I}} and [[angular frequency|angular speed]] {{mvar|ω}} measured in radians per second.<ref name="Worthington">{{cite book
 |last1 = Worthington
 |first1 = Arthur M.
 |title = Dynamics of Rotation
 |publisher = Longmans, Green and Co., London
 |date=1906
 |url=https://books.google.com/books?id=eScXAAAAYAAJ|page= 21|via= Google books
}}</ref>
<math display="block" qid=Q161254>L = I\omega.</math>

Unlike mass, which depends only on amount of matter, moment of inertia depends also on the position of the axis of rotation and the distribution of the matter. Unlike linear velocity, which does not depend upon the choice of origin, orbital angular velocity is always measured with respect to a fixed origin. Therefore, strictly speaking, {{mvar|L}} should be referred to as the angular momentum ''relative to that center''.<ref name="Taylor90">
{{cite book
 |last1 = Taylor
 |first1 = John R.
 |title = Classical Mechanics
 |url = https://archive.org/details/classicalmechani00jrta
 |url-access = limited
 |publisher = University Science Books, Mill Valley, CA
 |date=2005
 |isbn=978-1-891389-22-1
 |page=[https://archive.org/details/classicalmechani00jrta/page/n104 90]
}}</ref>

In the case of circular motion of a single particle, we can use <math>I = r^2m</math> and <math>\omega = {v}/{r}</math> to expand angular momentum as <math>L = r^2 m \cdot {v}/{r},</math> reducing to:
<math display="block">L = rmv,</math>

the product of the [[radius]] of rotation {{mvar|r}} and the linear momentum of the particle <math>p = mv</math>, where <math>v= r\omega</math> is the [[Speed#Tangential speed|linear (tangential) speed]].

This simple analysis can also apply to non-circular motion if one uses the component of the motion [[perpendicular]] to the [[radius vector]]:
<math display="block">L = rmv_\perp,</math>
where <math>v_\perp = v\sin(\theta)</math> is the perpendicular component of the motion. Expanding, <math>L = rmv\sin(\theta),</math> rearranging, <math>L = r\sin(\theta)mv,</math> and reducing, angular momentum can also be expressed,
<math display="block">L = r_\perp mv,</math>
where <math>r_\perp = r\sin(\theta)</math> is the length of the [[torque#Moment arm formula|''moment arm'']], a line dropped perpendicularly from the origin onto the path of the particle. It is this definition, {{math|(length of moment arm) × (linear momentum)}}, to which the term ''moment of momentum'' refers.<ref name="Dadourian">{{cite book
|last1 = Dadourian
|first1 = H. M.
|title = Analytical Mechanics for Students of Physics and Engineering
|publisher = D. Van Nostrand Company, New York
|date=1913
|url=https://books.google.com/books?id=1b3VAAAAMAAJ|page= 266|via= Google books
}}</ref>