The mass of an object $ M$ is also sometimes called its inertia, because it is harder to get a heavy object moving than a light one. In rotations, the moment of inertia $ I$ is a measure of how hard it is to rotate an object. It is harder to rotate an object with a larger $ I$. $ \omega$ is the angular velocity of the object, the amount of radians per second with which the object is rotating.

The inertia of one point of mass $ m_i$, with a distance $ r_i$ to the axis of rotation, adds to the moment of inertia of an object:

$\displaystyle I_i = m_i r_i^2$ (16)

An object can be considered as formed by very many material points, so the total moment of inertia is:

$\displaystyle I = \sum_i m_i r_i^2$ (17)

Instead of a material point we can also take a small volume $ \d V$ with a density $ \rho$ situated at a distance $ r_i$ from the axis of rotation. We can then write down:

$\displaystyle m_i = \rho \d V \quad \Rightarrow \quad I_i = \rho r_i^2 \d V$ (18)

The total moment of inertia is then:

$\displaystyle I = \sum_i \rho r_i^2 \d V$ (19)

In the limit of an infinetisimal volume $ \d V$ the sum can be replaced by an integral:

$\displaystyle I = \int_V \rho r^2 \d V$ (20)

where the subscript $ V$ below the integral denotes the integration over the entire volume $ V$ of the object.