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.
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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 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 Np [point particles]. Each of the Np point particles (of mass mi where i runs from 1 to Np) will obey Newton's 2nd Law:
\begin
[ \sum_
^{N_{\rm f,i}} \vec
_
= m_
\vec
_
] \end
where Nf,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:
\begin
[ \sum_
^{N_{\rm f,i}} \vec
_
\times \vec
_
= m_
\vec
_
\times \vec
_
]\end
We can rewrite this using the definition of the angular acceleration and the [torque]:
\begin
[ \sum_
^{N_{\rm f,i}} \tau_
= m_
r_
^
\alpha ] \end
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.
Implementing a sum over the particles that make up the body then gives:
\begin
[ \sum_
^{N_{\rm p}} \sum_
^{N_{\rm f,i}} \tau_
= \alpha \sum_
^{N_{\rm p}} m_
r_
^
]\end