Editing Moment of inertia (section)

== Definition ==
The '''moment of inertia''' is defined as the product of mass of section and the square of the distance between the reference axis and the [[centroid]] of the section. [[File:Cup of Russia 2010 - Yuko Kawaguti (2).jpg|thumb|right|upright|Spinning figure skaters can reduce their moment of inertia by pulling in their arms, allowing them to spin faster due to [[conservation of angular momentum]].]]
[[File:25. Ротационен стол.ogv|thumb|right|Video of rotating chair experiment, illustrating moment of inertia. When the spinning professor pulls his arms, his moment of inertia decreases; to conserve angular momentum, his angular velocity increases.]]

The moment of inertia {{mvar|I}} is also defined as the ratio of the net [[angular momentum]] {{mvar|L}} of a system to its [[angular velocity]] {{mvar|ω}} around a principal axis,<ref name="Winn" /><ref name="Fullerton" /> that is
<math qid=Q165618 display="block">I = \frac{L}{\omega}.</math>

If the angular momentum of a system is constant, then as the moment of inertia gets smaller, the angular velocity must increase. This occurs when spinning [[figure skating spins|figure skater]]s pull in their outstretched arms or [[Diving (sport)|diver]]s curl their bodies into a [[Diving (sport)#Positions|tuck position]] during a dive, to spin faster.<ref name="Winn">
{{cite book
 | last1  = Winn | first1 = Will 
 | title  = Introduction to Understandable Physics: Volume I - Mechanics
 | publisher = AuthorHouse
 | date   = 2010
 | pages  = 10.10
 | url    = https://books.google.com/books?id=NH8m7j9V0cUC&q=%22ice+skater%22+%22moment+of+inertia&pg=SA10-PA10
 | isbn   = 978-1449063337
}}</ref><ref name="Fullerton">
{{cite book
 | last1  = Fullerton | first1 = Dan 
 | title  = Honors Physics Essentials
 | publisher = Silly Beagle Productions
 | date   = 2011
 | pages  = 142–143
 | url    = https://books.google.com/books?id=8XmF2dy-9YYC&q=%22ice+skater%22+%22moment+of+inertia&pg=PA143
 | isbn   = 978-0983563334
}}</ref><ref name="Wolfram">
{{cite web
 | last = Wolfram | first = Stephen
 | title = Spinning Ice Skater
 | website = Wolfram Demonstrations Project
 | publisher = Mathematica, Inc.
 | date = 2014
 | url = http://demonstrations.wolfram.com/SpinningIceSkater/
 | access-date = September 30, 2014
}}</ref><ref name="Hokin">
{{cite web
 | last = Hokin | first = Samuel
 | title = Figure Skating Spins
 | work = The Physics of Everyday Stuff
 | date = 2014
 | url = http://www.bsharp.org/physics/spins
 | access-date = September 30, 2014
}}</ref><ref name="Breithaupt" >
{{cite book
 | last1  = Breithaupt | first1 = Jim 
 | title  = New Understanding Physics for Advanced Level
 | publisher = Nelson Thomas
 | date   = 2000
 | pages  = 64
 | url    = https://books.google.com/books?id=r8I1gyNNKnoC&q=%22ice+skater%22+%22moment+of+inertia&pg=PT73
 | isbn   = 0748743146
}}</ref><ref name="Crowell">
{{cite book
 | last1  = Crowell
 | first1 = Benjamin 
 | title  = Conservation Laws
 | publisher = Light and Matter
 | date   = 2003
 | pages  = [https://archive.org/details/conservationlaws0000crow/page/107 107]
 | url    = https://archive.org/details/conservationlaws0000crow
 | url-access = registration
 | quote  = ice skater conservation of angular momentum.
 | isbn   = 0970467028
}}</ref>

If the shape of the body does not change, then its moment of inertia appears in [[rotation around a fixed axis|Newton's law of motion]] as the ratio of an [[torque|applied torque]] {{mvar|τ}} on a body to the [[angular acceleration]] {{mvar|α}} around a principal axis, that is<ref name="Lerner">
{{cite book
 | last = Lerner
 | first = Lawrence S.
 | title = Physics for Scientists and Engineers
 | publisher = Jones and Bartlett
 | date = 1996
 | url = https://books.google.com/books?id=kJOnAvimS44C
 | archive-url= 
 | archive-date=
 | doi = 
 | id = 
 | isbn = 0867204796
 | mr = 
 | zbl = 
 | jfm =}}</ref>{{rp|279}}<ref name="Tipler">{{cite book
 | last1  = Tipler
 | first1 = Paul A.
 | title  = Physics for Scientists and Engineers, Vol. 1: Mechanics, Oscillations and Waves, Thermodynamics
 | publisher = Macmillan
 | date   = 1999
 | url    = https://books.google.com/books?id=U9lkAkTdAosC&q=skater+%22conservation+of+angular+momentum&pg=PA304
 | isbn   = 1572594918
}}</ref>{{rp|261, eq.9-19}}
<math qid=Q48103 display="block">\tau = I \alpha.</math>

For a [[simple pendulum]], this definition yields a formula for the moment of inertia {{mvar|I}} in terms of the mass {{mvar|m}} of the pendulum and its distance {{mvar|r}} from the pivot point as,
<math display="block">I = mr^2.</math>

Thus, the moment of inertia of the pendulum depends on both the mass {{mvar|m}} of a body and its geometry, or shape, as defined by the distance {{mvar|r}} to the axis of rotation.

This simple formula generalizes to define moment of inertia for an arbitrarily shaped body as the sum of all the elemental point masses {{math|''dm''}} each multiplied by the square of its perpendicular distance {{mvar|r}} to an axis {{mvar|k}}. An arbitrary object's moment of inertia thus depends on the spatial distribution of its mass.

In general, given an object of mass {{mvar|m}}, an effective radius {{mvar|k}} can be defined, dependent on a particular axis of rotation, with such a value that its moment of inertia around the axis is
<math display="block">I = m k^2,</math>
where {{mvar|k}} is known as the [[radius of gyration]] around the axis.