Thread: moment of inertia.

Damn near everything I've seen online for moments of inertia, as a reference, only includes moments of inertia that have the axis of rotation either through one end of the object, or through the center of mass. I'm trying to figure out how to find the moment of inertia for an object that is rotating about an axis that is outside of the body of rotation, like say, one that is attached to an axle by a massless, infinitely thin and rigid rod. What's the easiest way to calculate this for, say, a uniform cylinder or sphere where the center of mass is some distance away from the axis of rotation?

Originally Posted by Arcane_Mathematician

Damn near everything I've seen online for moments of inertia, as a reference, only includes moments of inertia that have the axis of rotation either through one end of the object, or through the center of mass. I'm trying to figure out how to find the moment of inertia for an object that is rotating about an axis that is outside of the body of rotation, like say, one that is attached to an axle by a massless, infinitely thin and rigid rod. What's the easiest way to calculate this for, say, a uniform cylinder or sphere where the center of mass is some distance away from the axis of rotation?

The moment of inertial is defined about some given axis. Just calculare the moment about the axis that is of interest to you. I may involve a nasty integral, but it is the same sort of integral that arises for the cases that you mention.

The only reason that the axes that you find easily in the literature are the ones that you do find there is that they are the cases that are usually of interest.

Okay. So I'm looking to solve an integral such as this:

with r being a radius vector, rho being the mass density, d(r) being the distance to the axis of rotation, and the integral evaluated over the volume of the object. I'm not sure what the radius vector is exactly, but I imagine it's in reference either to the axis of rotation or the center of mass of the object?

This was taken from Wiki's page on the Moment of Inertia. I'm not entirely clear on what dV(r) represents in the expression, could I get some help with that?

Originally Posted by Arcane_Mathematician

I'm not sure what the radius vector is exactly, but I imagine it's in reference either to the axis of rotation or the center of mass of the object?

No. It is the distance to the axis in question from the point of integration.