Kinematics

2014-01-23T14:04:55Z

208.74.117.9: /* Kinematic pairs */

{{infobox unit
| name = Radians per second squared
| standard = [[SI derived unit]]
| quantity = Angular acceleration
| symbol = rad/s{{sup|2}}
| symbol2 = rad⋅s<sup>−2</sup>
}}
{{Classical mechanics|cTopic=Fundamental concepts}}
'''Angular acceleration''' is the rate of change of [[angular velocity]]. In [[SI]] units, it is measured in [[radian]]s per [[second]] squared (rad/s{{sup|2}}), and is usually denoted by the Greek letter [[alpha (letter)|alpha]] (''α'').<ref> http://theory.uwinnipeg.ca/physics/circ/node3.html </ref>

== Mathematical definition ==

The angular acceleration can be defined as either:

:<math>{\alpha} = \frac{{d\omega}}{dt} = \frac{d^2{\theta}}{dt^2}</math> , or <br>

:<math>{\alpha} = \frac{a_T}{r}</math> ,

where <math>{\omega}</math> is the [[angular velocity]], <math>a_T</math> is the linear [[tangential acceleration]], and <math>r</math>, (usually defined as the radius of the circular path of which a point moving along), is the distance from the origin of the [[coordinate system]] that defines <math>\theta</math> and <math>\omega</math> to the point of interest.

== Equations of motion ==
For two-dimensional [[rotation|rotational motion]] (constant <math>\hat L</math>), [[Newton's laws of motion#Newton's second law|Newton's second law]] can be adapted to describe the relation between [[torque]] and angular acceleration:

:<math>{\tau} = I\ {\alpha}</math> ,
where <math>{\tau}</math> is the total torque exerted on the body, and <math>I</math> is the [[Moment_of_inertia|mass moment of inertia]] of the body.

===Constant acceleration===
For all constant values of the torque, <math>{\tau}</math>, of an object, the angular acceleration will also be constant. For this special case of constant angular acceleration, the above equation will produce a definitive, constant value for the angular acceleration:

:<math>{\alpha} = \frac{\tau}{I}.</math>

===Non-constant acceleration===

For any non-constant torque, the angular acceleration of an object will change with time. The equation becomes a differential equation instead of a constant value. This [[differential equation]] is known as the equation of motion of the system and can completely describe the motion of the object. It is also the best way to calculate the angular velocity.

== See also ==

* [[Angular momentum]]
* [[Angular speed]]
* [[Angular velocity]]
* [[Rotation]]
* [[Spin (physics)|Spin]]

==References==
{{reflist}}

[[Category:Physical quantities]]
[[Category:Acceleration]]

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Kinematics