Editing Angular momentum (section)

=== Angular momentum and torque ===
{{See also|Torque#Relationship with the angular momentum}}
[[Newton's laws of motion#Newton's second law|Newton's second law of motion]] can be expressed mathematically,
<math display="block">\mathbf{F} = m\mathbf{a},</math>
or [[force]] = [[mass]] × [[acceleration]]. The rotational equivalent for point particles may be derived as follows:
<math display="block">\mathbf{L} = I\boldsymbol{\omega}</math>
which means that the torque (i.e. the time [[derivative]] of the angular momentum) is
<math display="block" qid=Q48103>\boldsymbol{\tau} = \frac{dI}{dt}\boldsymbol{\omega} + I\frac{d\boldsymbol{\omega}}{dt}. </math>

Because the moment of inertia is <math>mr^2</math>, it follows that <math>\frac{dI}{dt} = 2mr\frac{dr}{dt} = 2rp_{||}</math>, and <math>\frac{d\mathbf{L}}{dt} = I\frac{d\boldsymbol{\omega}}{dt} + 2rp_{||}\boldsymbol{\omega},</math> which, reduces to
<math display="block">\boldsymbol{\tau} = I\boldsymbol{\alpha} + 2rp_{||}\boldsymbol{\omega}.</math>
This is the rotational analog of Newton's second law. Note that the torque is not necessarily proportional or parallel to the angular acceleration (as one might expect). The reason for this is that the moment of inertia of a particle can change with time, something that cannot occur for ordinary mass.