Moment of inertia

The moment of inertia of an object is the resistance of the object to changes in its rotational state of motion. For a particle of mass \(m\), located at a distance of \(R\) from the axis, the moment of inertia is defined as

\[I := m R^2.\]

This definition already shows us that moment of inertia is always dependent on the choice of the axis (through the distance from the axis \(R\)). Taken about a different axis, the moment of inertia will be different.

Moment of inertia is a scalar quantity.

The SI unit of moment of inertia is the kilogramme metre squared:

\[\left[I\right] = 1\,\mathrm{kg} \mathrm{m}^2.\]

For an extended object, determining the moment of inertia is more complex. We have to determine the moment of inertia of parts of the object and add them up. Let us divide the rigid body into small, homogeneous parts of equal volume \(\Delta V\), each with mass \(\Delta m_i\) and at a distance \(R_i\) from the axis of rotation. Let us denote the density at the location of the \(i\)th small part by

\[\rho_i = \rho\left(x_i, y_i, z_i\right),\]

where \(x_i\), \(y_i\) and \(z_i\) are the Cartesian coordinates of the \(i\)th small part. Mass is related to density by

\[\Delta m_i = \rho_i \Delta V.\]

Thus the moment of inertia of the \(i\)th small part is
\[I_i = R_i^2 \Delta m_i = R_i^2 \rho_i \Delta V.\]

The moment of inertia of the whole rigid body is the sum of the individual moments:

\[I = \sum\limits_i\left(R_i^2 \rho_i \Delta V\right).\]

To be more precise, we must use an integral instead of the sum:
\[I = \lim\limits_{\Delta V \to 0}\sum\limits_i{\left(R_i^2 \rho_i \Delta V\right)} = \int\limits_V R^2 \rho \mathrm{d} V = \int\int\int R^2(x, y, z)\rho(x, y, z)\mathrm{d} x \mathrm{d} y \mathrm{d} z.\]

This notation means that we must perform three integrations with respect to the individual coordinates.