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View Full Version : Re: More on inertia? [and Centrifugal Force]

Paolo De Leva
12-17-2000, 04:32 AM
Dear BIOMCH-L subscribers,

I believe it is important to insist a little bit more, with the
kindness, friendship and respect that you all deserve for the high
"understanding" the dynamics of a rotating object. In this message, I will
try to be more explicit than in my message dated December 13 2000 about the
same point, in
the hope that we all can eventually agree about it (or more easily
understand it's wrong and agree on the contrary).

I will briefly comment on a
statement written by Michael Schwartz in his recent contribution to
BIOMCH-L:

> Finally, d'Alembert's principle makes clear the
> exact origin and nature of the different apparent
> forces arising from moving reference frames...
> ...Well that's my \$0.02....

Besides that I believe Michael's contribution was worth much more than
0.02 \$ for its rich and stimulating contents (I'll ask the library of my
wouldn't say that D'Alembert's principle makes clear the origin of the
inertial forces (and torques, as far as I know; didn't D'Alembert deal with
inertial couples as well, or someone else did it?).

In my personal opinion,
D'Alembert's principle just helps you to figure out the value of these
forces. Nothing more. Just merely their value, which you can blindly compute
even not knowing at all their origin (and that's the danger of D'Alembert's
principle). If you want to understand their origin, you need to use Newton's
first law, and of course consider the kinematics observed from an inertial
frame.
For instance, to understand the origin of the apparent force which, on
our planet, tilts toward east the trajectory of a falling body (making it
not perfectly vertical), or tilts towards west the trajectory of a
projectile shoot by a gun pointing vertically upward, you need to consider
the absolute velocity that the falling object or the projectile have
initially, which:

- depends on the rotation of the earth, (which you can see only from a
"fixed" reference frame, according to Newton's approach)
- is not zero (as you would say observing the phenomenon in a non-inertial
reference frame attached to the earth, in which D'Alembert's principle is
applied)
- is different from the velocity of all the points vertically above or below
it (again due to the rotation of our planet)..

With kind regards,

Paolo de LEVA

Ist. Universitario di Scienze motorie
Biomechanics Laboratory
P. Lauro De Bosis, 6
00194 ROME - ITALY

----- Original Message -----
From: Michael Schwartz
To:
Sent: Wednesday, December 13, 2000 6:09 PM
Subject: More on inertia?

> I cannot resist adding my thoughts (perhaps there
> should be an entire listserv dedicated to inertial
> forces).
>
> My thoughts are not unique, but rather adapted
> (stolen) from the brilliant work
>
> "The Variational Principles of Mechanics" by
> Cornelius Lanczos (Dover Inc., New York).
>
> I would strongly urge anyone involved in mechanics
> to read as much of this as possible. It is fairly
> dense material but gives amazing insight. In Ch.
> IV, Lanczos deals exclusively with d'Alembert's
> Principle. When I first saw this, I was
> confused. I had always thought (been taught) that
> this "principle" was merely algebraic shuffling.
> However, it is noted by Lanczos that "it is
> exactly this apparent triviality which makes
> d'Alembert's principle such an ingenious invention
> and at the same time so open to distortion and
> misunderstanding.".
>
> This current "debate" is a great example of how
> prophetic Lanczos was!!!
>
> The most important point is that by transforming a
> non-equilibrium problem to an equilibrium problem
> by the addition of the inertial forces, the
> Principal of Virtual Work now holds (the virtual
> work of all forces vanishes). This then means
> that the PVW and all of it's consequuences can be
> directly applied to a "dynamic" problem.
>
> One of these consequences, related to gait
> analysis, is to allow the use of "kinematical
> variables", that is, velocities that are not the
> derivatives of actual position coordinates. The
> example cited by Lanczos is the spin of a top
> about its axis of symmetry, however one could
> extend this to think of the angular velocities of
> Euler angles.
>
> Finally, d'Alembert's principle makes clear the
> exact origin and nature of the different apparent
> forces arising from moving reference frames
> whether in translation, rotation or change in rate
> or direction of rotation (mass * omega_dot X R),
> which Lanczos dubs the "Euler force".
>
> Well that's my \$0.02 as we say on this side of the
> pond. Thanks to all contributors for the
> stimulating discussion.
>
> --
> Michael Schwartz, Ph.D.
>
> Director of Bioengineering Research
> Gillette Children's Specialty Healthcare
>
> Assistant Professor
> Orthopaedic Surgery, Biomedical Engineering
> University of Minnesota
>
> Phone:(651)229-3929 Fax:(651)229-3867
>
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