Editing Force (section)

=== Rotation and torque ===
[[File:Torque animation.gif|frame|right|Relationship between force (''F''), torque (''τ''), and [[angular momentum|momentum]] vectors (''p'' and ''L'') in a rotating system.]]
{{main|Torque}}
Forces that cause extended objects to rotate are associated with [[torque]]s. Mathematically, the torque of a force <math> \mathbf{F}</math> is defined relative to an arbitrary reference point as the [[cross product]]:
<math display="block" qid="Q48103">\boldsymbol\tau = \mathbf{r} \times \mathbf{F},</math>
where <math> \mathbf{r}</math> is the [[position vector]] of the force application point relative to the reference point.<ref name="openstax-university-physics" />{{rp|497}}

Torque is the rotation equivalent of force in the same way that [[angle]] is the rotational equivalent for [[position (vector)|position]], [[angular velocity]] for [[velocity]], and [[angular momentum]] for [[momentum]]. As a consequence of Newton's first law of motion, there exists [[rotational inertia]] that ensures that all bodies maintain their angular momentum unless acted upon by an unbalanced torque. Likewise, Newton's second law of motion can be used to derive an analogous equation for the instantaneous [[angular acceleration]] of the rigid body:
<math display="block">\boldsymbol\tau = I\boldsymbol\alpha,</math>
where
* <math>I</math> is the [[moment of inertia]] of the body
* <math> \boldsymbol\alpha</math> is the angular acceleration of the body.<ref name="openstax-university-physics"/>{{rp|502}}

This provides a definition for the moment of inertia, which is the rotational equivalent for mass. In more advanced treatments of mechanics, where the rotation over a time interval is described, the moment of inertia must be substituted by the [[Moment of inertia tensor|tensor]] that, when properly analyzed, fully determines the characteristics of rotations including [[precession]] and [[nutation]].<ref name=":0" />{{Rp|pages=96–113}}

Equivalently, the differential form of Newton's second law provides an alternative definition of torque:<ref>{{cite web |last=Nave |first=Carl Rod |title=Newton's 2nd Law: Rotation |work=HyperPhysics |publisher=University of Guelph |url=http://hyperphysics.phy-astr.gsu.edu/HBASE/n2r.html |access-date=2013-10-28}}</ref>
<math display="block">\boldsymbol\tau = \frac{\mathrm{d}\mathbf{L}}{\mathrm{dt}},</math>
where <math> \mathbf{L}</math> is the angular momentum of the particle.

Newton's third law of motion requires that all objects exerting torques themselves experience equal and opposite torques,<ref>{{cite web |last=Fitzpatrick |first=Richard |title=Newton's third law of motion |date=2007-01-07 |url=http://farside.ph.utexas.edu/teaching/336k/lectures/node26.html |access-date=2008-01-04}}</ref> and therefore also directly implies the [[conservation of angular momentum]] for closed systems that experience rotations and [[revolution (geometry)|revolution]]s through the action of internal torques.