what is the proof of moment of inertia of a solid rod
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The integral means you take the contribution r2dm of each of these tiny bit of matter to the total moment of inertia. The position (the value of r) of your element in a cylinder varies from the inner radius to the outer radius. If your cylinder is not hollow, this means your inner radius is zero.
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Moment of Inertia: Rod. The moment of inertia about the end of the rod can be calculated directly or obtained from the center of mass expression by use of the Parallel axis theorem. I = kg m². If the thickness is not negligible, then the expression for I of a cylinder about its end can be used.
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