Kinematics: Difference between revisions

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{{infobox unit
| name    = Radians per second squared
| standard = [[SI derived unit]]
| quantity = Angular acceleration
| symbol  = rad/s{{sup|2}}
| symbol2  = rad&sdot;s<sup>&minus;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]]

Latest revision as of 17:36, 1 January 2015

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