The Moments of Inertia

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Definition

Moment of inertia, denoted by the letter I, is another name for rotational inertia. It is associated with an object that is rotating about its center, or an axis. The moment of inertia for any rotating objects must be specified with respect to a chosen axis of rotation due to the varying distance r. The moment of inertia corresponds to mass for translational/linear motion. [1] [2]

A Mathematical Model

The point mass model of the moment of inertia.

The moment of inertia is the sum of mass times the square of perpendicular distance to the rotation axis, that is [math]\displaystyle{ I=\Sigma mr^2 }[/math]for all point mass components. This relationship is the basis for all other moments of inertia since any object can be built up from a collection of point masses. [3] Note that I has units of mass multiplied by distance squared (kg⋅m2), as we might expect from its definition.

Moments of Inertia of Different Shapes

Some common uniform-density solids whose moments of inertia are known.

Hoop

The moment of inertia of a hoop or thin hollow cylinder of negligible thickness about its central axis is a straightforward extension of the moment of inertia of a point mass since all of the mass is at the same distance R from the central axis.[4]

Shpere

The expression for the moment of inertia of a sphere can be developed by summing the moments of infintesmally thin disks about the z axis. The moment of inertia of a thin disk is [5]

Thin Rod

The moment of inertia calculation for a uniform rod involves expressing any mass element in terms of a distance element dr along the rod. To perform the integral, it is necessary to express eveything in the integral in terms of one variable, in this case the length variable r. Since the total length L has mass M, then M/L is the proportion of mass to length and the mass element can be expressed as shown. Integrating from -L/2 to +L/2 from the center includes the entire rod. The integral is of polynomial type:

Examples

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