In our study of rotational dynamics we skipped over exactly how to calculate the rotational inertia of a solid body. The process for calculating this quantity is quite complicated, and requires quite a bit of calculus. Thus we devote a section to calculating this quantity.

Consider a small section of a rod, a radius r from the axis of rotation, and with a mass
*δm*, as shown below:

Figure %: A small piece of mass on a rod being rotated about an axis.

I | = | r_{k}^{2}δm_{k} | |

= | r^{2}dm |

This integral equation is the basic equation for the moment of inertia of a solid body.

Even with this equation, it is quite difficult to calculate the moment of inertia of a solid body. We will go through an example to show how it is done. Let us simply return to the example of the solid rod of length L, and mass M, rotated about its center, as shown below.

Figure %: A small piece of mass, dm, of length dx on a rod being rotated about an axis.

I | = | r^{2}dm | |

= | x^{2}(ρAdx) | ||

= | x^{2}(Adx) | ||

= | x^{2}dx |

Thus we now have an integral we can evaluate. We simply need to determine the limits. If we denote the axis of rotation to be at

I | = | x^{2}dx | |

= | []_{-L/2}^{L/2} | ||

= | ML^{2} |

This is the equation for the moment of inertia of a thin rod, and it agrees with measured values.

In general, the moment of inertia of a solid body varies with *MR*^{2}, where R is the
measure of the radius, or length of a given object. To find the exact value of the moment
of inertia, however, the complicated calculus is required.

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