Question on moment of inertia tensor

[Johnny5]

I want to understand the following formula:

 

t=I a

 

Where t denotes torque, a denotes angular acceleration, and I denotes the moment of inertia tensor.

 

I've been doing a great deal of reading, looking for someone who understands it more clearly than any other, and I've had no luck.

 

I really really want to understand "moment of inertia tensor"

 

Is there anyone who has a deep understanding of it, who can shed some light.

 

Even historical information is welcome. Like who developed the formula, when why how.

 

I have those kinds of questions.

 

Also, I think this thread might ultimately be quite valuable to others, who don't understand that formula either.

 

I would like the thread to be moderated, remain on topic, and be professional.

 

I would like it to proceed from the simple to the complex.

 

I will not ask dumb questions, and am prepared to do some independent study. I currently have gathered several books which discuss it to varying degrees, but again... I haven't read a single person's coverage of the matter, to the point where I can say, wow that guy's good.

 

There are some brilliant people that frequent this site, and I know this.

 

I want to understand precession especially.

 

The boomerang question also got me back to this question.

 

I've never understood the moment of inertia tensor, so I can rectify that now.

 

Umm... lets see... there was something else I wanted to say...

 

Oh yeah...

 

Yesterday, I was thinking about how I would proceed to teach the topic, if i already understood it, and I came up with this...

 

I would buy a tiny toy spinning top.

 

Then, to my pretend student, I could repeatedly perform a simple experiment.

 

Set the top spinning on the floor, and have them watch the precession of the axis of rotation.

 

Then, slowly I would develop a mathematical model of the toy.

 

I could mark places on the top, where imaginary axes go.

 

I could say, there is a frame, attached to the top, whose axes spin with it.

 

The axes of this spinning frame pass through the points I've made on this toy top, and one of the axes of this three dimensional coordinate system is permanently the axis of rotation.

 

So we need to come up with formulas which predict the precession of this top, which motion we can see by repeatedly performing the experiment.

 

I might then talk about density, mass per unit volume of the top, then again perhaps not.

 

I might touch on the integral calculus, and how it is used to "discuss" the location of the center of mass of the top.

 

But what I really have no idea about, is how to get to "moment of inertia tensor" from just watching the thing spin.

 

So that's what I really want to do.

 

Thank you

 

PS: Now here is wolfram on moment of inertia

 

Here is a site that is beyond understandable: Rotation of rigid bodies

 

 

Now in something I was recently reading, it was Euler who extended Newton's laws to rotational motion. I don't know how true that is.

 

There are "rotational analog" formulas, to Newton's linear ones.

 

I know the formulas, here is a link:

 

Rotational Dynamics

 

At the site above, they say "fixed axis" I'm not sure what that means.

 

Here is a site that looks good: Rotational Dynamics

 

And here is exactly what I was looking for, a side by side comparison of the formulas:

 

Hyperphysics on Moment Of Inertia

 

I think the first question to answer, is why doesn't the spinning top fall to the ground, it stays up in a manner of speaking. If you nudge it, it pushes back, to remain spinning.

 

And also, there is the ice skater.

 

She pulls her arms in, and spins faster.

 

I know many of you know how to handle this topic.

 

I am hoping for a clear presentation, something i cannot seem to find in my books.

 

I recently went back to a classical mechanics book, written by umm

 

I can't remember his name, but it doesn't matter.

 

He glossed over things. his treatement of the subject was ultimately incoherent. I could tell he understood it on some level, yet well here I am asking for someone who knows how to teach to try to explain.

 

I really think the presentation needs to be based upon some real experiment with a spinning top, and not be overly mathematical. however, you have to go from a spinning top, to tensors, so eventually things get abstract. I'm ok with that.

 

but if someone were to come to me right now, and ask me why doesn't a spinning top fall to the ground, I would be unable to impress them with my answer.

 

That's going to change.

[Johnny5]

I am going to start myself off, generate my own knowledge as it were.

 

Here is a definition for "moment of inertia"

 

Definition: moment of inertia

 

Moment of inertia quantifies the resistance of a physical object to angular acceleration. Moment of inertia is to rotational motion as mass is to linear motion.

 

So at least that is a start, if anyone here objects to this definition, please tell me why. PS, completely disregard the theory of relativity. I would like a total non relativistic treatment of the subject.

 

 

The reason being, is that my toy top is not spinning at the speed of light.

 

Now, here is something they say at the site, which I now have permanently memorized:

 

the greater the concentration of material away from the object's centroid, the larger the moment of inertia.

 

The above is quite understandable. Every object has its own unique center of mass, which can be described in its own rest frame. This was explained in the mechanics book i was talking about, but the author did a poor job explaining it.

 

To paraphrase... the greater the concentration of material away from the objects center of mass, the larger is its resistance to being made to spin faster. (resistance to angular acceleration)

 

Right now I am fixated on a bicycle wheel for some reason.

 

This site also mentions the parallel axis theorem which i dont know. I vaguely remember it, but I surely did not memorize it.

 

As I recall, once I accept the universality of conservation of angular momentum, the rest follows.

 

Perhaps someone will take that route.

 

Once latex is working again, I am going to go heavy on the mathematical treatement of this.

 

Definition: L = r X P

 

r is the moment arm, F the applied force. All three are vectors.

 

 

Ok here is a question, who came up with this formula, it is like so totally non-intuitive...

 

I = moment of inertia = S m_i (r_i)²

 

People just don't know things like this by magic.

 

Here is the thing which I apparently keep missing:

 

However, in the general case of an object being rotated about an arbitrary axis, the moment of inertia becomes a tensor, such that the angular momentum need not be parallel to the angular velocity. The definition of the moment of inertia tensor is very similiar to that above, except that it is now expressed as a matrix:

 

This site is good.

 

Ok they kind of start off with this:

 

Angular momentum in classical mechanics

 

The traditional mathematical definition of the angular momentum of a particle about some origin is:

 

L=R X P

 

 

 

where

 

L is the angular momentum of the particle,

 

r is the position of the particle expressed as a displacement vector from the origin

 

p is the linear momentum of the particle.

 

If a system consists of several particles, the total angular momentum about an origin can be obtained by adding (or integrating) all the angular momenta of the constituent particles. Angular momentum can also be calculated by multiplying the square of the distance to the point of rotation, the mass of the particle and the angular velocity.

 

That's a place to start.

 

Now, they aren't clear where this origin is to be located though.