Page History

{excerpt}An object's moment of inertia is a measure of the effort required to change that object's rotational velocity about a specified axis of rotation. {excerpt}

||Page Contents||
|{toc:style=none|indent=10px}|

h2. Mathematical Definition

For an introductory course, it is sufficient to consider the definition of the moment of inertia of a [rigid body] executing pure rotation (no tranlation relative to the axis) about an [axis of rotation] that maintains a fixed distance from that [rigid body's|rigid body] center of mass.  The importance of this statement is that every point in the body will *maintain a fixed distance from the [axis of rotation]*.  This condition is specified so that the moment of inertia of the body remains constant.  

Under this condition, we can quickly derive the form and the utility of the moment of inertia by considering the [rigid body] to be a collection of _N_~p~ [point particles].  Each of the _N_~p~ point particles (of mass _m_~i~ where i runs from 1 to _N_~p~) will obey [Newton's 2nd Law|Newton's Second Law]:

{latex}\begin{large}\[ \sum_{j=1}^{N_{\rm f,i}} \vec{F}_{i,j} = m_{i}\vec{a}_{i} \] \end{large}{latex}

where _N_~f,i~ is the number of forces acting on the ith particle.

Taking the cross product of each side of this equation with respect to the radial distance from the axis of rotation:

{latex}\begin{large}\[ \sum_{j=1}^{N_{\rm f,i}} \vec{r}_{i} \times \vec{F}_{i,j} = m_{i} \vec{r}_{i}\times \vec{a}_{i} \]\end{large}{latex}

We can rewrite this using the definition of the [angular acceleration] and the [torque]:

{latex}\begin{large} \[ \sum_{j=1}^{N_{\rm f,i}} \tau_{i,j} = m_{i} r_{i}^{2} \alpha \] \end{large}{latex}

{note}Note that for a [rigid body] that is undergoing pure rotation about a certain axis (recall our assumption), all particles will have the same angular acceleration.{note}

Implementing a sum over the particles that make up the body then gives:

{latex}\begin{large}\[ \sum_{i=1}^{N_{\rm p}} \sum_{j=1}^{N_{\rm f,i}} \tau_{i,j} = \alpha \sum_{i=1}^{N_{\rm p}} m_{i}r_{i}^{2}\]\end{large}{latex}