section_path: Concepts derived from force/Rotation and torque
section_name: Rotation and torque
paragraph_text: Rotation and torque
  
  Forces that cause extended objects to rotate are associated with torques. Mathematically, the torque of a force F {\displaystyle \mathbf {F} } is defined relative to an arbitrary reference point as the cross product: τ = r × F, {\displaystyle {\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F},} where r {\displaystyle \mathbf {r} } is the position vector of the force application point relative to the reference point.
  
  Torque is the rotation equivalent of force in the same way that angle is the rotational equivalent for 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: τ = I α, {\displaystyle {\boldsymbol {\tau }}=I{\boldsymbol {\alpha }},} where
  
  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 tensor that, when properly analyzed, fully determines the characteristics of rotations including precession and nutation.
  
  Equivalently, the differential form of Newton's Second Law provides an alternative definition of torque: τ = d L d t, {\displaystyle {\boldsymbol {\tau }}={\frac {\mathrm {d} \mathbf {L} }{\mathrm {dt} }},} where L {\displaystyle \mathbf {L} } 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, and therefore also directly implies the conservation of angular momentum for closed systems that experience rotations and revolutions through the action of internal torques.