Editing Angular acceleration (section)

=== Relation to torque ===

The net ''[[torque]]'' on a point particle is defined to be the pseudovector

: <math qid=Q48103>\boldsymbol{\tau} = \mathbf r \times \mathbf F,</math>

where <math>\mathbf F</math> is the net force on the particle.<ref name="ref3">{{cite book |last1=Singh |first1=Sunil K. |title=Torque |url=https://cnx.org/contents/MymQBhVV@175.14:JOsDHAfQ@4/Torque |publisher=Rice University |ref=3}}</ref>

Torque is the rotational analogue of force: it induces change in the rotational state of a system, just as force induces change in the translational state of a system. As force on a particle is connected to acceleration by the equation <math qid=Q11402>\mathbf F = m\mathbf a</math>, one may write a similar equation connecting torque on a particle to angular acceleration, though this relation is necessarily more complicated.<ref>{{cite book|last1=Mashood|first1=K.K.|url=http://www.hbcse.tifr.res.in/research-development/ph.d.-theses/thesis-mashoodkk.pdf |title=Development and evaluation of a concept inventory in rotational kinematics |publisher=Tata Institute of Fundamental Research, Mumbai|pages=52–54|ref=4}}</ref>

First, substituting <math>\mathbf F = m\mathbf a</math> into the above equation for torque, one gets

: <math>\boldsymbol{\tau} = m\left(\mathbf r\times \mathbf a\right) = mr^2 \left(\frac{\mathbf r\times \mathbf a}{r^2}\right).</math>

From the previous section:

: <math>\boldsymbol{\alpha}=\frac{\mathbf r\times \mathbf a}{r^2}-\frac{2}{r} \frac{dr}{dt}\boldsymbol{\omega},</math>

where <math>\boldsymbol{\alpha}</math> is orbital angular acceleration and <math>\boldsymbol{\omega}</math> is orbital angular velocity. Therefore:

: <math>\boldsymbol{\tau} = mr^2 \left(\boldsymbol{\alpha}+\frac{2}{r} \frac{dr}{dt}\boldsymbol{\omega}\right) 
=mr^2 \boldsymbol{\alpha}+2mr\frac{dr}{dt}\boldsymbol{\omega}. </math>

In the special case of constant distance <math>r</math> of the particle from the origin (<math>\tfrac{ dr } {dt} = 0</math>), the second term in the above equation vanishes and the above equation simplifies to

: <math>\boldsymbol{\tau} = mr^2\boldsymbol{\alpha},</math>

which can be interpreted as a "rotational analogue" to <math>\mathbf F = m\mathbf a</math>, where the quantity <math>mr^2</math> (known as the [[moment of inertia]] of the particle) plays the role of the mass <math>m</math>. However, unlike <math>\mathbf F = m\mathbf a</math>, this equation does ''not'' apply to an arbitrary trajectory, only to a trajectory contained within a spherical shell about the origin.