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Rotations and torque

Relationship between force (F), torque (τ), and momentum vectors (p and L) in a rotating system.

{{#invoke:main|main}} Forces that cause extended objects to rotate are associated with torques. Mathematically, the torque of a force <math>\scriptstyle \vec{F}</math> is defined relative to an arbitrary reference point as the cross-product:

<math>\vec{\tau} = \vec{r} \times \vec{F}</math>

where

<math>\scriptstyle \vec{r}</math> 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:

<math>\vec{\tau} = I\vec{\alpha}</math>

where

<math>I</math> is the moment of inertia of the body
<math>\scriptstyle \vec{\alpha}</math> is the angular acceleration of the body.

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:

<math>\vec{\tau} = \frac{\mathrm{d}\vec{L}}{\mathrm{dt}},</math><ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref> where <math>\scriptstyle \vec{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>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> and therefore also directly implies the conservation of angular momentum for closed systems that experience rotations and revolutions through the action of internal torques.

Centripetal force

{{#invoke:main|main}} For an object accelerating in circular motion, the unbalanced force acting on the object equals:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<math>\vec{F} = - \frac{mv^2 \hat{r}}{r}</math>

where <math>m</math> is the mass of the object, <math>v</math> is the velocity of the object and <math>r</math> is the distance to the center of the circular path and <math>\scriptstyle \hat{r}</math> is the unit vector pointing in the radial direction outwards from the center. This means that the unbalanced centripetal force felt by any object is always directed toward the center of the curving path. Such forces act perpendicular to the velocity vector associated with the motion of an object, and therefore do not change the speed of the object (magnitude of the velocity), but only the direction of the velocity vector. The unbalanced force that accelerates an object can be resolved into a component that is perpendicular to the path, and one that is tangential to the path. This yields both the tangential force, which accelerates the object by either slowing it down or speeding it up, and the radial (centripetal) force, which changes its direction.<ref name=FeynmanVol1 /><ref name=Kleppner />


Force sections
Intro  Development of the concept  Pre-Newtonian concepts  Newtonian mechanics  Special theory of relativity  Descriptions  Fundamental forces  Non-fundamental forces  Rotations and torque  Kinematic integrals  Potential energy  Units of measurement  Force measurement  See also  Notes  References  Further reading  External links  

Rotations and torque
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