What is Inertia

[Don A. Gilmore]

----- Original Message ----- 
From: "Mark Davidson" <mark.davidson@mindspring.com>
To: <caut@ptg.org>
Sent: Wednesday, December 24, 2003 10:50 AM
Subject: What is Inertia


> 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!?).

Actually, this is an elementary branch of mechanics called "statics".  A net
force of zero is the same as no force at all.  If someone pushes on you from
the left with 10 lbs and someone from the right with 10 lbs, you won't go
anywhere.  If you rearrange the F=ma equation you get

a = F/m

If F is zero, so is a.  The result is not independent of mass; the mass is
still in the equation, it's just getting multiplied by zero.

> 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)."
>
> 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.

Be careful using momentum for anything.  It is a more "imaginary" concept
dealing with impulse and conservation of momentum in one direction, or about
one axis.  There isn't even a standard letter for it in physics, it's just
referred to as "mv".

> 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.

You are correct.  The first component (mr^2/4) becomes neglible as the
length becomes much larger in comparison to the shank diameter and the
second component (mL^2/3) gets larger as a result.

> 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?

I never meant to imply that calculus should be used in determining the
m.o.i. of piano action parts.  You would simply break it down into simple
components like cylinders, disks, prisms, etc. and then use offset equations
and add them together.  As you pointed out, the approximation would probably
be adequate for what we're trying to accomplish here.

Actually, in the engineering world, if a part is so complex that it is
difficult to calculate a property easily and accurately, we just measure it.
There actually is such a thing as a "moment of inertia gauge".  It simply
spins the object at a prescribed acceleration and measures the torque
produced.   That's about as accurate as you can get.

Don A. Gilmore
Mechanical Engineer
Kansas City
>
> -Mark
>
>
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