Moment of Inertia
Moment of inertia is the name given to rotational
inertia, the rotational analog of mass for linear motion. It appears in the
relationships for the dynamics of rotational motion. The moment of inertia must
be specified with respect to a chosen axis of rotation. For a point mass, the
moment of inertia is just the mass times the square of perpendicular distance
to the rotation axis, I = mr^2. That point mass relationship becomes the basis
for all other moments of inertia since any object can be built up from a
collection of point masses.
Moment of inertia is defined with respect to a specific rotation axis. The moment of inertia of a point mass with respect to an axis is defined as the product of the mass times the distance from the axis squared. The moment of inertia of any extended object is built up from that basic definition. The general form of the moment of inertia involves an integral.
Moment of Inertia: General Form
The moment of inertia of
an object/body involves a continuous distribution of mass at a continually altering
(varying) distance from any axis-of-rotation (rotational axis), the calculation
of Moments of Inertia generally incorporates calculus, the discipline of
mathematics that can handle such continuous variables. Since the moment of
inertia (I) of a point mass is expressed by formula:
Then, the moment of inertia contribution by an
infinitesimal (very small) mass element dm has the same form. This kind
of mass element is called a differential element of mass and its moment of
inertia is given by the expression below:
Note that the differential element of moment of inertia dI must always be defined with respect to a specific rotation axis. The sum over all these mass elements is called an integral over the mass.
Usually, the mass element dm will be expressed in terms of the geometry of the object, so that the integration can be carried out over the object as a whole (for example, over a long uniform rod).
Moment of Inertia || Mass Moment of Inertia
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