Editing Moment of inertia (section)

=== Simple pendulum ===
Mathematically, the moment of inertia of a simple pendulum is the ratio of the torque due to gravity about the pivot of a pendulum to its angular acceleration about that pivot point. For a simple pendulum, this is found to be the product of the mass of the particle <math>m</math> with the square of its distance <math>r</math> to the pivot, that is
<math display="block">I = mr^2.</math>

This can be shown as follows:

The force of gravity on the mass of a simple pendulum generates a torque <math>\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}</math> around the axis perpendicular to the plane of the pendulum movement. Here <math>\mathbf{r}</math> is the distance vector from the torque axis to the pendulum center of mass, and <math>\mathbf{F}</math> is the net force on the mass. Associated with this torque is an [[angular acceleration]], <math>\boldsymbol{\alpha}</math>, of the string and mass around this axis. Since the mass is constrained to a circle the tangential acceleration of the mass is <math>\mathbf{a} = \boldsymbol{\alpha} \times \mathbf{r}</math>. Since <math>\mathbf F = m \mathbf a</math> the torque equation becomes:
<math display="block">\begin{align}
  \boldsymbol{\tau}
    &= \mathbf{r} \times \mathbf{F}
     = \mathbf{r} \times (m \boldsymbol{\alpha} \times \mathbf{r}) \\
    &= m \left(\left(\mathbf{r} \cdot \mathbf{r}\right) \boldsymbol{\alpha} - \left(\mathbf{r} \cdot \boldsymbol{\alpha}\right) \mathbf{r}\right) \\
    &= mr^2 \boldsymbol{\alpha}
     = I\alpha \mathbf{\hat{k}},
\end{align}</math>
where <math>\mathbf{\hat{k}}</math> is a unit vector perpendicular to the plane of the pendulum. (The second to last step uses the [[Triple product#Vector triple product|vector triple product expansion]] with the perpendicularity of <math>\boldsymbol{\alpha}</math> and <math>\mathbf{r}</math>.) The quantity <math>I = mr^2</math> is the ''moment of inertia'' of this single mass around the pivot point.

The quantity <math>I = mr^2</math> also appears in the [[angular momentum]] of a simple pendulum, which is calculated from the velocity <math>\mathbf{v} = \boldsymbol{\omega} \times \mathbf{r}</math> of the pendulum mass around the pivot, where <math>\boldsymbol{\omega}</math> is the [[angular velocity]] of the mass about the pivot point. This angular momentum is given by
<math display="block">\begin{align}
  \mathbf{L}
    &= \mathbf{r} \times \mathbf{p}
     = \mathbf{r} \times \left(m\boldsymbol{\omega} \times \mathbf{r}\right) \\
    & = m\left(\left(\mathbf{r} \cdot \mathbf{r}\right)\boldsymbol{\omega} - \left(\mathbf{r} \cdot \boldsymbol{\omega}\right)\mathbf{r}\right) \\
    &= mr^2 \boldsymbol{\omega}
     = I\omega\mathbf{\hat{k}},
\end{align}</math>
using a similar derivation to the previous equation.

Similarly, the kinetic energy of the pendulum mass is defined by the velocity of the pendulum around the pivot to yield
<math display="block">E_\text{K} = \frac{1}{2} m \mathbf{v} \cdot \mathbf{v} = \frac{1}{2} \left(mr^2\right)\omega^2 = \frac{1}{2}I\omega^2.</math>

This shows that the quantity <math>I = mr^2</math> is how mass combines with the shape of a body to define rotational inertia. The moment of inertia of an arbitrarily shaped body is the sum of the values <math>mr^2</math> for all of the elements of mass in the body.