In physics you deal with units. However you decide or choose which conditions you include in your theory, if you can't "recover" the physics--an actual path "s" through space, over a finite interval of time "t", there is no hope. If you "abstract away" the mass, or anything physical for some reason (why it would be or who to, are different problems), then you have to put it back to recover observational results.
In vector spaces unit vectors can replace any basis set of vectors, in that space. Force and momentum are both vectors, but are not in the same vector space(s).
In mechanics of rigid bodies in motion, mass is distributed into a volume with a shape.
One of the curious things about the moment of inertia I, is that it's related to a fixed (by the geometry and a chosen axis of rotation), radius, the radius of gyration. This is another of those geometric things about the gravitational/inertial interactions.
The radius, call it a thing defined at a boundary, a geometric condition or constraint. has a number of small volume elements somewhere along it; the problem of finding these two geometric conditions--I and the radius of gyration--is a calculus problem, finding the volume of revolution. Unless you don't have a nice ring or flat disc, you have to get geometric another way. Finding the radius of gyration gives you the moment of inertia because of a geometric relation. The mass of a rigid body with an homogenous density of matter is fixed for any small volume element you sum over.
But you need to have this mass-density relation, calculus otherwise will let you deal with a small ring as if it's a 1-torus with a lineal mass density. You recover observable physics by finding the inertial moments along this abstract 1-torus or "thin" ring. So a disc is a sum of small, concentric rings, so I for a ring is basic, or a foundation for the next composite shape, or distribution of matter, the disc, which is topologically equivalent to a bowl, which is . . .
In vector spaces unit vectors can replace any basis set of vectors, in that space. Force and momentum are both vectors, but are not in the same vector space(s).
In mechanics of rigid bodies in motion, mass is distributed into a volume with a shape.
One of the curious things about the moment of inertia I, is that it's related to a fixed (by the geometry and a chosen axis of rotation), radius, the radius of gyration. This is another of those geometric things about the gravitational/inertial interactions.
The radius, call it a thing defined at a boundary, a geometric condition or constraint. has a number of small volume elements somewhere along it; the problem of finding these two geometric conditions--I and the radius of gyration--is a calculus problem, finding the volume of revolution. Unless you don't have a nice ring or flat disc, you have to get geometric another way. Finding the radius of gyration gives you the moment of inertia because of a geometric relation. The mass of a rigid body with an homogenous density of matter is fixed for any small volume element you sum over.
But you need to have this mass-density relation, calculus otherwise will let you deal with a small ring as if it's a 1-torus with a lineal mass density. You recover observable physics by finding the inertial moments along this abstract 1-torus or "thin" ring. So a disc is a sum of small, concentric rings, so I for a ring is basic, or a foundation for the next composite shape, or distribution of matter, the disc, which is topologically equivalent to a bowl, which is . . .
