Just as mass traveling in straight line has inertia that tends to keep velocity constant, mass rotating around axis has inertia {moment of inertia}| {inertia moment} that tends to keep angular velocity constant.
mass
Inertia depends directly on mass. Moment of inertia substitutes for mass when quantities use angular velocity instead of velocity. When mass rotates around rotation center at radius r, tangential momentum pt is mass m times tangential velocity vt: pt = m * vt. Angular momentum L is moment of inertia I times angular velocity w: L = I * w. Tangential velocity vt varies directly with angular velocity w: vt = w * r. Tangential momentum pt varies directly with angular momentum L: L = pt * r. Moment of inertia I depends on mass m and radius r: L = I * w = pt * r = m * w * r * r = (m * r^2) * w. Moment of inertia I is mass m times square of distance r from axis or point: I = m * r^2.
summation
Masses have volume, so object points have different radii from rotation center. Total moment of inertia is sum of moments of inertia at each radius.
Thin-ring moment of inertia equals total mass m times square of distance r from ring center to ring middle: m * r^2.
Disk or cylinder moment of inertia equals half total mass m times square of distance r from disk or cylinder center to outer edge: 0.5 * m * r^2.
Pipe or doughnut moment of inertia equals half total mass m times sum of squares of distances from pipe or doughnut center to inner edge ri and outer edge ro: 0.5 * m * (ri^2 + ro^2).
radius
Objects with moment of inertia around rotation center have moment of inertia around any axis parallel to rotation axis. New moment of inertia Inew is old moment of inertia Iold plus total object mass m times square of distance d between axes: Mnew = Mold + m * d^2.
Physical Sciences>Physics>Dynamics>Rotation
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Date Modified: 2022.0224