Re: MOMENT OF INTERTIA

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

On Thu, 17 Dec 1998, David Abineri wrote:

> If one starts with a hoop and thinks about gradually filling in the
> inside of the hoop with mass until it eventually looks like a disk, one
> is certainly increasing the moment of intertia of the object. The effact
> of the above transformation is to produce an object which will roll ever
> faster down a given incline. I can show this easily mathematically but
> is there an intuitive way of explaining why this happens? My students
> understand the mathematics but seem to need a non mathematical argument
> too.

 From the context of the rest of your message I assume that by "gradually
filling in the inside of the hoop with mass until it eventually looks like
a disk" you mean that you fill it in *from* the outside *toward* the
center.  In that case the object rolls down the incline with ever greater
acceleration as you add mass because the ratio of the radius of gyration
to the rolling radius becomes monotonically smaller.  (Recall that the
radius of gyration is simply sqrt(I/M).)  Indeed the acceleration is given
by

                     a_frictionless
   a =  -----------------------------------------
        1 + (radius of gyration/rolling radius)^2

where a_frictionless is the acceleration that an object would have if it
slid down the same incline without friction (or if its radius of gyration
were 0!)

As far as a nonmathematical argument goes, I would look to the
mathematical result itself (and its derivation) for clues.  What it says
to me is that distributing the mass of a rolling object relatively closer
to the axis makes it relatively easier to roll.  It also says that we
should not allow ourselves to get hung up on the absolute amount of mass
in the object since it contributes proportionally to the inertia of the
object *and* to the net force on the object--a lesson that we learned (?)
the first time we dropped a crumpled piece of paper next to a steel ball.

Note that this is the same "intuitive" (?) explanation that we use to
understand why solid spheres of any size and any mass, beat discs of any
size and any mass, which beat hollow spheres ... , which beat hoops ...,
when rolling from rest down an incline.

John

Fundamental Laws of Physics are apparent truths that we do not understand.
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