Steiner's Theorem: Moment of Inertia Around Axis

[dx]

I didn't understand the following argument :

"Suppose we have an object, and we want to find its moment of inertia around some axis. That means we want the inertia needed to carry it by rotation about that axis. Now if we support the object on pivots at the center of mass, so that the object does not turn as it rotates about the axis ( because there is no torque on it from inertial effects, and therefore it will not turn when we start moving it), then the forces needed to swing it around are the same as though all the mass were concentrated at the center of mass, and the moment of inertia would be simply I_{1} = MR_{CM}^2 where R_{CM} is the distance from the axis to the center of mass. But of course that is not the right formula for the moment of inertia of an object which is really being rotated as it revolves, because not only is the center of it moving in a circle, which would contribute an amount I_{1} to the moment of inertia, but also we must turn it about its center of mass. So it is not unreasonable that we must add I_{1} to the moment of inertia I_{c} about the center of mass. So, its is a good guess that the total moment of inertia about any axis will be

I = I_{c} + MR_{CM}^2"

 

[Tide]

That's simply a way of "motivating" the parallel axis theorem. Basically, it is saying that, aside from spinning about a principal axis, an object rotating about a parallel axis probably behaves as though all its mass were concentrated at a point. There are, of course, more rigorous ways of arriving at that conclusion.

 

[dx]

Yea, I know you can do it rigorously, but this particular argument I can't understand. For instance, what does this sentence mean : "that means we want the inertia needed to carry it by rotation about that axis". Isnt inertia the resistance to angular acceleration? or is it some measure of how much rotation it can give a body. Why would inertia be 'needed'? I am sorry for the naive comments, but rotation is a new topic for me. I would really appreciate it if you could reconstruct the whole argument and present it in an expanded form. Thank you.

 

[Tide]

I think it is poorly worded. You may be better off thinking of it as meaning something like "we want the inertia of the object rotating about that axis."

 

[dx]

and what about this ? "because there is no torque on it from inertial effects, and therefore it will not turn when we start moving it"

 

[Tide]

That just means that, in this contrived situation, the object is suspended at its center of mass so that spinning about a principle axis will not change.

 

[dx]

whats a principal axis, will not change what?

 

[Tide]

A principal axis is one for which the moment of inertia is a minimum, e.g. an axis of symmetry. I was referring to the spin not changing.

 

[dx]

ok, thanks for the help.