Re: [Phys-L] inertia ... or not

[John Denker <jsd@av8n.com>]

On 05/31/2012 04:04 PM, Kirk Bailey wrote:
>  In my mind, when introducing inertia,
> one must mention that inertia is the resistance to a change in the state of
> motion, not resistance to acceleration (because rotating objects are always
> accelerating).  

Bingo.  That definitely answers the question that I asked.  As is 
typical of phys-l, the answer is more sophisticated than the original
question.

Let me say it in my own words, to see if I understand it.  According to
this viewpoint, we can say:

    The scalar inertia = m 
                       = ∫ dm

    The tensor inertia = I 
                       = ∫  r^2  − r ⊗ r   dm
                         ∫ ⟨r|r⟩ − |r⟩⟨r|  dm 

where ⊗ is one way of denoting the outer product (i.e. tensor product).

Both the scalar inertia and the tensor inertia are examples of inertia.
That's not the only way of looking at things, and I'm not 100% convinced
it's the optimal way, but it is elegant and self-consistent.

=========

Here's another way of looking at things:  One could argue that the plain
old scalar mass *is* the inertia:

    Inertia = mass 
         = zeroth moment of mass 
         = m 
         = ∫ dm

In contrast:

    So-called "moment of inertia" = shorthand for *second* moment of inertia
           = second moment of mass 
           = I
           = ∫  r^2  − r ⊗ r dm
           = ∫ ⟨r|r⟩ − |r⟩⟨r|  dm 

> From this point of view, the moment of inertia is not "the" rotational inertia,
but rather the rotational /analog/ of inertia i.e. the rotational /analog/ of
mass ... just as in an electrical LC oscillator we can say the L and the C are
/analogous/ to a mass and a spring (or vice versa) without requiring them to
be literally "the" mass and "the" spring.

I haven't figured out whether this viewpoint is in any way better or worse
than the previous argument.  Maybe they're both equally viable.  I need to
think about it some more.

=========

At the tactical level, in the introductory course, I still reckon it makes 
sense to start by introducing mass, directly.  This is how I was taught.
Maybe I'm stuck in a rut, but I just don't see the advantage of introducing
two concepts when one will do.  Even if we decide there is a more general 
notion of inertia, it seems to me that it could wait until much later.

==================================================

Returning to the level of sophisticated physics:

As for BC's point about gravitational mass versus inertial mass:  You have
to be very careful how you say that.  According to general relativity, the
mass is exactly the inertia and only approximately the source-term for the
gravitational field.  More generally the stress-energy tensor is the source-
term.  It reduces to mass in the static limit, but that's not the whole story, 
just as in the case of Maxwell's equations, electrostatics is important but
it's not the whole story.

==========

Pursuing a different tangent:  I would remark that the statement of Newton's 
third law in the usual introductory physics texts is incorrect or at best 
incomplete, because it doesn't say anything about the /point of application/ 
of the forces.

As such, it cannot distinguish between two forces that cancel, producing no
effect whatsoever, and two forces with a lever arm producing a torque.

  The second law has similar issues, but let's not go there right now.

To say the same thing another way:  The third law is often stated in a way 
that is tantamount to conservation of momentum.  That's fine as far as it 
goes, but we also need conservation of /angular/ momentum.

  FWIW Newton got this right in the Principia.  The textbooks in question
  have dumbed it down.

There is a something of a dilemma here:
 -- You can't teach everything at once.  You have to start somewhere.  You 
  have to make some quick & dirty simplifications the first time through.
 -- The students tend to take the book literally.  They have no way of
  knowing what's a quick & dirty simplification and what's a deep law
  of nature.

I just wish the textbook authors would be more fastidious about labeling
the approximations as such.  It wouldn't kill them to say that such-and-such
equation applies in simple cases, and then refer to a more complete discussion
in a later chapter.