Editing Moment of inertia (section)

== Examples ==
{{See also|List of moments of inertia}}

=== 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.

=== Compound pendulums ===
[[File:Mendenhall gravimeter pendulums.jpg|thumb|left|Pendulums used in Mendenhall [[gravimeter]] apparatus, from 1897 scientific journal. The portable gravimeter developed in 1890 by Thomas C. Mendenhall provided the most accurate relative measurements of the local gravitational field of the Earth.]]
A [[compound pendulum]] is a body formed from an assembly of particles of continuous shape that rotates rigidly around a pivot. Its moment of inertia is the sum of the moments of inertia of each of the particles that it is composed of.<ref name="B-Paul">
{{cite book
 | last =Paul
 | first =Burton
 | title =Kinematics and Dynamics of Planar Machinery
 | publisher =Prentice Hall 
 | date =June 1979
 | isbn =978-0135160626
}}</ref><ref name=Resnick>
{{cite book
 | last1=Halliday | first1=David
 | last2=Resnick | first2=Robert
 | last3=Walker | first3=Jearl
 | title=Fundamentals of physics|year=2005|publisher=Wiley
 | location=Hoboken, NJ
 | isbn=9780471216438
 | edition=7th
}}</ref>{{rp|pp=395–396}}<ref>
{{cite book
 | last=French
 | first=A.P.
 | title=Vibrations and waves
 | year=1971
 | publisher=CRC Press
 | location=Boca Raton, FL
 | isbn=9780748744473
}}</ref>{{rp|pp=51–53}} The [[resonance|natural]] [[angular frequency|frequency]] (<math>\omega_\text{n}</math>) of a compound pendulum depends on its moment of inertia, <math>I_P</math>,
<math display="block">\omega_\text{n} = \sqrt{\frac{mgr}{I_P}},</math>
where <math>m</math> is the mass of the object, <math>g</math> is local acceleration of gravity, and <math>r</math> is the distance from the pivot point to the center of mass of the object. Measuring this frequency of oscillation over small angular displacements provides an effective way of measuring moment of inertia of a body.<ref name="Uicker"/>{{rp|pp=516–517}}

Thus, to determine the moment of inertia of the body, simply suspend it from a convenient pivot point <math>P</math> so that it swings freely in a plane perpendicular to the direction of the desired moment of inertia, then measure its natural frequency or period of oscillation (<math>t</math>), to obtain
<math display="block">I_P = \frac{mgr}{\omega_\text{n}^2} = \frac{mgrt^2}{4\pi^2},</math>
where <math>t</math> is the period (duration) of oscillation (usually averaged over multiple periods).

==== Center of oscillation ====
A simple pendulum that has the same natural frequency as a compound pendulum defines the length <math>L</math> from the pivot to a point called the [[center of oscillation]] of the compound pendulum. This point also corresponds to the [[center of percussion]]. The length <math>L</math> is determined from the formula,
<math display="block">\omega_\text{n} = \sqrt{\frac{g}{L}} = \sqrt{\frac{mgr}{I_P}},</math>
or
<math display="block">L = \frac{g}{\omega_\text{n}^2} = \frac{I_P}{mr}.</math>

The [[seconds pendulum]], which provides the "tick" and "tock" of a grandfather clock, takes one second to swing from side-to-side. This is a period of two seconds, or a natural frequency of <math>\pi \ \mathrm{rad/s}</math> for the pendulum. In this case, the distance to the center of oscillation, <math>L</math>, can be computed to be
<math display="block">L = \frac{g}{\omega_\text{n}^2} \approx \frac{9.81 \ \mathrm{m/s^2}}{(3.14 \ \mathrm{rad/s})^2} \approx 0.99 \ \mathrm{m}.</math>

Notice that the distance to the center of oscillation of the seconds pendulum must be adjusted to accommodate different values for the local acceleration of gravity. [[Kater's pendulum]] is a compound pendulum that uses this property to measure the local acceleration of gravity, and is called a [[gravimeter]].