Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: MOMENT OF INERTIA



Hi Mervin,
Thanks for your further comments.
The points you make about the definition of the system for
the person pushing off the wall and whether or not zero work
is done are, I believe, covered by what you've re-read in
the relevant pages in Arnold's book.

I am interested in your use of the term "mechanism". To what does
that
refer, and what insight does that give us in solving the problem.

Mechanism: the machinery, or the agencies or means, by
which a particular effort is produced or a purpose is
accomplished.

So I see the mechanism as the set-up by which a process
occurs: what the person does to get across tp the other side
of the room - pushes against the wall; what happens to the
moving box to bring it to a halt - sliding on a rough
surface; what arrangement makes the hoop roll down the
incline.

Then I can ask questions about the process: what are the
changes in energy; what are the forces involved; what are
the time-lines for the accelerations, velocities and momenta
of the system and/or parts of the system during the process;
what transfers of energy in or out of the system by work or
heat are involved; what psuedo-work is done on the center of
mass of the system?

I guess the use of the term mechanism is an idiosyncrasy of
mine: one of these things that seem blindingly obvious to
oneself but... I think its use helps set up the situation
but I may well be astray on this.


For the hoop rolling down the incline (which is close to
where this thread started) there is a transfer of energy
from the gravitational potential energy of the system of
hoop and earth to translational and rotational kinetic
energy of the hoop. The mechanisms are the gravitational
force between the hoop and the earth and the contact force
between the incline surface and the hoop.

The contact force is a constraint force. It doesn't do any
work but, together with the gravitational force, it
determines the motion. First it stops the hoop from
falling. Second it supplies the torque that leads to the
rotation. That's why I refer to it as (part of) the
mechanism for the process to occur. The process gives us
the motion we observe and then, acting like accountants, we
can do our sums to finds how much energy is in each of the
different forms.

I interpret your reference to the contact force as meaning the
force
normal to the surface of contact. Since that force is through the
center
of rotation, it cannot be responsible for the torque that produces
the
rotation.

No, I'm taking the contact force as all the contact force,
both parallel and normal components. If there is no
parallel component, which would be true if the incline were
perfectly smooth, then, as you point out the contact force
would be through the center of rotation, the torque would be
zero and the hoop would slide down the incline. For a rough
incline the full contact force is not normal to the surface
and a non-zero torque results. Certainly we can (and
usually do) separate the contact force back to its parallel
and normal components to calculate the torque from the
parallel, frictional component. However, and again I may be
idiosyncratic, I'd rather think of the contact force as a
single force or interaction between the hoop and the
incline.

Brian McInnes