What is Inertia

[Mark Davidson]

Richard,

RicB wrote

>What I'd like to see at this point is that since Don, Sarah, Mark, and
>Jim all are people we all rely on for physics insights, and because they
>all present clearly different definitions of this concept,,, that these
>four all bang this one through until they arrive at a common definiton
>for us.

...

Well, after digging back into my physics book (which doesn't define things
anywhere as nicely as a math book does), I'm inclined to agree with Don too.
Newtons' first law says:

"Every object persists in its state of rest or uniform motion in a straight
line unless it is compelled to change that state by forces impressed on it."

Keep in mind that Newton was working on an explanation of orbiting planets,
which do NOT move in straight lines (implying invisible forces).  The
concept of gravitational force reaching out into space was pretty wild.
Also the idea that things keep moving without pushing on them was
counter-intuitive.  You push on something, it stops.  Push it down a hill,
it stops at the bottom.  Things don't just keep going, right?

My physics book did note that the first law is really a special case of the
second.  I.e., when net force is zero, then F = M A implies no change in
speed, regardless of mass (hence this property is independent of mass!?).

But the first law does beg the question - If I have to push on it to
accelerate it, then how hard do I have to push for a given acceleration.
The same force pushing on different weight objects results in different
accelerations.  This is basically the concept of mass.  Mass is very
intuitive because it is proportional to weight (force of gravity on mass)
which we deal with every day.

Newton's 2nd law is first stated in terms of momentum, however:

"Force is equal to the change in momentum (mV) per change in time. (For a
constant mass, force equals mass times acceleration. F = m a)."

>grin.... NOW I will state my own position... tentatively...ok ?? :)
>Seems to me that Don is correct... except I have a hard time
>understanding or accepting that "one  object cannot have more "inertia"
>than another". If this is true then either inertia is a constant, or
>inertia is just plain undefined... as in divideing by zero more or less.
>So I lean towards Sarah and Mark. But I want to see you 4 hashing this
>out so we can past the problem.... as clearly any discussion about
>action mechanics on this list is going to be rather meaningless unless
>we can agree on what terms like inertia mean.

So, I still think that conceptually you are thinking about momentum - a
moving object will push on you if it runs into you. (Force = change in
momentum/time).  .

A couple of other notes I'd like to throw in.

When I compared energy of a hammer and lead, I called it a "napkin sketch"
because I knew it was a very crude approximation, serving only to show that
something was amiss in terms of resistance at the key being due primarily to
key inertia, while most of the energy was going into the hammer.  I hold no
pretentions of that being a complete model of a piano action.  Just trying
to prevent a train wreck.

That said, sometimes approximations are simpler and work fine in practice.
For instance we have now seen two definitions of the moment of inertia of a
cylindrical stick rotating about its endpoint.

Don gives J = (1/4 * m * r^2) + (1/3 * m * L^2), where L is the length of
the shaft and r is its radius, for a cylindrical stick rotating about its
end. Frequently though, and I think we've seen it here, you will just see J
= 1/3 * m * L^2.  So what's going on here?

Well, Don's formula is technically the correct one (unfortunately hammer
shanks aren't cylindrical), because it takes into account radius.  But it
turns out that once the length is more than about 6.5 times the diameter,
the error is less than 0.5%.  Personally I think these kind of
approximations are fine in practice, but one should keep them in mind.  When
the book says a long thin rod, they mean a long thin rod.  When we talk
about the change (not total) in inertia due to adding a key lead, I am
pretty confident that we can use mass of the keylead and radius to its
center, and the change will be close to m*r^2.  No, it's not exact, but
again its close to within a small percentage, and gives us something that
most people can easily work with.

Finally, I would argue that while, yes, one can in theory figure out all
this stuff by doing lots of measuring and integrating, that is totally
impractical.  Every key has different dimensions, different lead placement,
etc, etc, and you just wouldn't ever finish.  I would much prefer a method
to directly measure or estimate with reasonable accuracy (as in the thin rod
example) the moment of inertia of key and hammer/shank.  You don't see the
guy down at the tire shop who balances your tires doing calculus, do you?

-Mark