What is Inertia

[Don A. Gilmore]

Hi Don (Rose):

Well, you're getting warmer.  I guess it's just all a matter of semantics.
Inertia is not an inherrent, quantifiable property of an object, it's an
effect, like the Doppler effect.  There are no units of "inertia"; one
object cannot have more "inertia" than another.  It can have more kinetic
energy, or momentum, or mass, or velocity, or indeed "moment" of inertia
than another object since those are measurable, quantifiable properties.

[I've been posting this info on several of the PTG discussion groups and
I've lost track of what I've posted where, so if you've already read some of
this I appologize.]

Moment of inertia has a misleading name since it uses the term "inertia".
It can be just thought of as the rotational equivalent of mass.  You can
even convert linear formulas to rotational ones simply by substituting angle
instead of distance, torque instead of force and moment of inertia instead
of mass.  Interesting, huh?

The trickiest one is the moment of inertia.  There is not a simple,
universal formula.  Moment of inertia is only equal to m * r^2 for a
"point-mass".  That is, a mass at a point in space, but with no size or
volume.  Obviously this is utter nonsense in the real world.  A real object
has some mass closer to the pivot and some farther away, so some points have
less m.o.i. and some have more.  As you break a mass up into smaller and
smaller pieces and add their individual moments of inertia together you get
a more and more accurate total.  To get the ultimate estimate takes adding
together an infinite number of infinitely small pieces of the object.  This
sounds impossible, but it is easy with integral calculus...and you will get
an *exact* result, not an approximation, believe it or not.

The good news is you don't normally have to use calculus every time you
figure moment of inertia.  For many classic geometric shapes there are
already formulas that have been calculated.  For example, for a cylindrical
rod (like a hammer shank) spinning about its end, the m.o.i. integrates to
the formula:

J = (1/4 * m * r^2) + (1/3 * m * L^2), where L is the length of the shaft
and r is its radius.

But for the same rod spinning about its middle the equation is

J = (1/4 * m * r^2) + (1/12 * m * L^2)

which is considerably different.  So now you can see that the m.o.i. is
dependent on the mass and shape of an object as well as the location of the
pivot point.

So forget about J=mr^2 since it is useless in its pure, unintegrated form.

Don A. Gilmore
Mechanical Engineer
Kansas City

----- Original Message ----- 
From: "Don" <pianotuna@accesscomm.ca>
To: "College and University Technicians" <caut@ptg.org>
Sent: Tuesday, December 23, 2003 2:40 PM
Subject: Re: What is Inertia


> Hi Rick
>
> Inertia is one thing. It sits there and is constant for a given mass.
>
> Moment of inertia is another.
>
> Apples and oranges--or better yet apples and apples swinging in a bag.
>
> At 02:25 AM 12/24/2003 +0100, you wrote:
> >
> >Hi Jim and Don.
> >
> >I cant help but thinking there is some significant difference between
> >the definitions of inertia each of you give below
>
> Regards,
> Don Rose, B.Mus., A.M.U.S., A.MUS., R.P.T.
>
> mailto:pianotuna@accesscomm.ca
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>
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