The Moments of Inertia

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Definition

Demonstrations of the moments of inertia about object's axis.

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 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 moment of inertia of a sphere can be expressed as the summation of moments of infinitesimally thin disks about the z axis. [5]

Cylinder

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These disks and cylinders all have moment of inertia [math]\displaystyle{ 1/2MR^2 }[/math]about their axes.

The moment of inertia of a solid cylinder can be calculated from the built up of moment of inertia of thin cylindrical shells. [6] Because only the perpendicular distances of atoms from the axis matter([math]\displaystyle{ r_\perp }[/math]), the moment of inertia for rotation about the axis of a long cylinder has exactly the same form as that of a disk. It means that the moment of inertia does not matter how long the cylinder is or how thick the disk is.

Other

The moments of inertia for different shapes can be calculated by applying integral calculus as shown in the calculation of moment of inertia for the sphere.

Examples

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