Editing Moment of inertia (section)

== Introduction ==
When a body is free to rotate around an axis, [[torque]] must be applied to change its [[angular momentum]]. The amount of torque needed to cause any given [[angular acceleration]] (the rate of change in [[angular velocity]]) is proportional to the moment of inertia of the body. Moments of inertia may be expressed in units of kilogram metre squared (kg·m<sup>2</sup>) in [[SI]] units and pound-foot-second squared (lbf·ft·s<sup>2</sup>) in [[imperial units|imperial]] or [[United States customary units|US]] units.

The moment of inertia plays the role in rotational kinetics that [[mass]] (inertia) plays in linear kinetics—both characterize the resistance of a body to changes in its motion. The moment of inertia depends on how mass is distributed around an axis of rotation, and will vary depending on the chosen axis. For a point-like mass, the moment of inertia about some axis is given by <math>mr^2</math>, where <math>r</math> is the distance of the point from the axis, and <math>m</math> is the mass. For an extended rigid body, the moment of inertia is just the sum of all the small pieces of mass multiplied by the square of their distances from the axis in rotation. For an extended body of a regular shape and uniform density, this summation sometimes produces a simple expression that depends on the dimensions, shape and total mass of the object.

In 1673, [[Christiaan Huygens]] introduced this parameter in his study of the oscillation of a body hanging from a pivot, known as a [[compound pendulum]].<ref name="mach">{{cite book |last=Mach |first=Ernst |title=The Science of Mechanics |year=1919 |pages=[https://archive.org/details/scienceofmechani005860mbp/page/n196 173]&ndash;187 |url=https://archive.org/details/scienceofmechani005860mbp |access-date=November 21, 2014}}</ref> The term ''moment of inertia'' ("momentum inertiae" in [[Latin]]) was introduced by [[Leonhard Euler]] in his book ''Theoria motus corporum solidorum seu rigidorum'' in 1765,<ref name="mach"/><ref name="Euler1730">{{Cite book |last=Euler |first=Leonhard |title=Theoria motus corporum solidorum seu rigidorum: Ex primis nostrae cognitionis principiis stabilita et ad omnes motus, qui in huiusmodi corpora cadere possunt, accommodata [The theory of motion of solid or rigid bodies: established from first principles of our knowledge and appropriate for all motions which can occur in such bodies.]|publisher= A. F. Röse|location= Rostock and Greifswald (Germany)|date= 1765|page= [https://archive.org/details/theoriamotuscor00eulegoog/page/n202 166]|url= https://archive.org/details/theoriamotuscor00eulegoog|language=la |isbn=978-1-4297-4281-8}} From page 166:  ''"Definitio 7.  422.  Momentum inertiae corporis respectu eujuspiam axis est summa omnium productorum, quae oriuntur, si singula corporis elementa per quadrata distantiarum suarum ab axe multiplicentur."'' (Definition 7.  422.  A body's moment of inertia with respect to any axis is the sum of all of the products, which arise, if the individual elements of the body are multiplied by the square of their distances from the axis.)</ref> and it is incorporated into [[Euler's laws#Euler's second law|Euler's second law]].

The natural frequency of oscillation of a compound pendulum is obtained from the ratio of the torque imposed by gravity on the mass of the pendulum to the resistance to acceleration defined by the moment of inertia. Comparison of this natural frequency to that of a simple pendulum consisting of a single point of mass provides a mathematical formulation for moment of inertia of an extended body.<ref name="Marion 1995">{{cite book |last1=Marion |first1=JB |last2=Thornton |first2=ST |year=1995 |title=Classical dynamics of particles & systems |edition=4th |publisher=Thomson |isbn=0-03-097302-3 |url-access=registration |url=https://archive.org/details/classicaldynamic00mari_0 }}</ref><ref name="Symon 1971">{{cite book |last=Symon |first=KR |year=1971 |title=Mechanics |edition=3rd |publisher=Addison-Wesley |isbn=0-201-07392-7}}</ref>

The moment of inertia also appears in [[angular momentum|momentum]], [[kinetic energy]], and in [[rigid body dynamics|Newton's laws of motion]] for a rigid body as a physical parameter that combines its shape and mass. There is an interesting difference in the way moment of inertia appears in planar and spatial movement. Planar movement has a single scalar that defines the moment of inertia, while for spatial movement the same calculations yield a 3&nbsp;×&nbsp;3 matrix of moments of inertia, called the inertia matrix or inertia tensor.<ref name="Tenenbaum 2004">{{cite book |last=Tenenbaum |first=RA |year=2004 |title=Fundamentals of Applied Dynamics |publisher=Springer |isbn=0-387-00887-X}}</ref><ref name="Kane">
{{cite book
 |first1=T. R.
 |last1=Kane
 |first2=D. A.
 |last2=Levinson
 |title=Dynamics, Theory and Applications
 |publisher=McGraw-Hill
 |location=New York
 |year=1985
}}</ref>

The moment of inertia of a rotating [[flywheel]] is used in a machine to resist variations in applied torque to smooth its rotational output. The moment of inertia of an airplane about its longitudinal, horizontal and vertical axes determine how steering forces on the control surfaces of its wings, elevators and rudder(s) affect the plane's motions in roll, pitch and yaw.