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Instructions
Identify the rotational axis. The moment of inertia of an object depends a lot on the axis around which it's spun. The moment of inertia of a dumbbell being spun around its axle, swung around by one end or flipped end-to-end about its middle are all different quantities.
As an example, take an object shaped like a capital Y. Assume the angle between the arms of the Y is 30 degrees and each section is of equal length, and say the object is rotating about a pin put right through the junction.
Find the form of the mass distribution of the object. You might, for example, have something that is equally dense all throughout, like a compact disk, or something like a dumbbell with circular weights that are denser than the barbell connecting them.
For the example, assume that the sections of the Y have no mass, but that each end is capped by a mass of M.
Multiply each mass by the square of its distance from the axis of rotation.
For the example problem, the distance from the axis of rotation for each mass is equal to the length of each section of the Y, call it R. The mass of each section is M, so multiplying each mass by the square of its distance gives M*R^2 for each of the three masses.
Add all the separate components from the last step.
For the example, the sum is M*R^2 + M*R^2 + M*R^2 = 3* M*R^2.