Editing Moment of inertia (section)

==== Angular momentum ====
The angular momentum vector for the planar movement of a rigid system of particles is given by<ref name="B-Paul"/><ref name="Uicker"/>
<math display="block">\begin{align}
  \mathbf{L} &= \sum_{i=1}^n m_i \Delta\mathbf{r}_i \times \mathbf{v}_i \\
             &= \sum_{i=1}^n m_i \,\Delta r_i\mathbf{\hat{e}}_i \times \left(\omega\, \Delta r_i\mathbf{\hat{t}}_i + \mathbf{V}\right) \\
             &= \left(\sum_{i=1}^n m_i \,\Delta r_i^2\right)\omega \mathbf{\hat{k}} +
                  \left(\sum_{i=1}^n m_i\,\Delta r_i\mathbf{\hat{e}}_i\right) \times \mathbf{V}.
\end{align}</math>

Use the [[center of mass]] <math>\mathbf{C}</math> as the reference point so
<math display="block">\begin{align}
    \Delta r_i \mathbf{\hat{e}}_i &= \mathbf{r}_i - \mathbf{C}, \\
  \sum_{i=1}^n m_i\,\Delta r_i \mathbf{\hat{e}}_i &= 0,
\end{align}</math>

and define the moment of inertia relative to the center of mass <math>I_\mathbf{C}</math> as
<math display="block">I_\mathbf{C} = \sum_{i} m_i\,\Delta r_i^2,</math>

then the equation for angular momentum simplifies to<ref name="Beer"/>{{rp|p=1028}}
<math display="block">\mathbf{L} = I_\mathbf{C} \omega \mathbf{\hat{k}}.</math>

The moment of inertia <math>I_\mathbf{C}</math> about an axis perpendicular to the movement of the rigid system and through the center of mass is known as the ''polar moment of inertia''. Specifically, it is the [[moment (physics)|second moment of mass]] with respect to the orthogonal distance from an axis (or pole).

For a given amount of angular momentum, a decrease in the moment of inertia results in an increase in the angular velocity. Figure skaters can change their moment of inertia by pulling in their arms. Thus, the angular velocity achieved by a skater with outstretched arms results in a greater angular velocity when the arms are pulled in, because of the reduced moment of inertia. A figure skater is not, however, a rigid body.