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Re: Force Sensors Must Measure Acceleration??



Regarding A. R. Marlow's poll about whether accelerometers measure
acceleration or force, I wonder if we can add a "neither" category, since if
I was to vote it would be "neither". It seems to me that such devices
(including bathroom scales, strain gauges, middle/inner ear sensory
structures, nerve endings in the feet and other body parts, etc.) respond to
some kind of elastic distortion on the part of the active sensing medium of
the apparatus. The net distortion is a cumulative effect of the local
elastic *strain* (either compressive, tensile, or shear) over some relatively
small region in the sensing medium. This distortion and strain is *neither
a force nor an acceleration*. To be blunt, they don't call them strain
gauges for nothing. To the extent that the sensing medium obeys Hook's law
(for sufficiently small strains for any solid) the local strain field tends
to be proportional to the local stress field applied to the sensing medium.
What produces the stress field in the sensing medium is a differential effect
of different parts of the sensing medium being subjected to different
external force densities. Since the medium relatively quickly equilibrates
its response to such locally differing force densities, and the medium as a
whole does not continue to experience ever more distortion, we see that
eventually the medium moves uniformly as a whole once a reading has been
obtained. This means that the sensing medium accelerates as a whole once the
measured distortion is established. The stress field is the medium's
*internal* forcing mechanism limiting the relative distortion which allows
each particle/parcel of the medium to have the same acceleration even though
they may be subject to differing *external* forces. The internal forces in
the medium in the form of the stress field makes up the difference so the
system can respond as a whole; for without such an internal stress, the
medium would continue to be torn apart as the different parts of it
accelerate and move with respect to each other in response to the differing
external forces present.

Now why would there be differing externally applied forces present? The
usual case is that there is some contact force on the sensing body which is
acting in a differential way on one part or side of the sensing body's
surface. For instance, at this moment the chair I am sitting on is pushing
up against my back side as I type this message. The chair does not uniformly
push against all parts of my body. (The air pressure does do this, but the
consequent overall bulk compressive strain which induces a bulk compressive
stress, i.e. internal pressure, in my body which equalizes the air pressure
is not considered a "distortion" and is a "normal" background configuation.)
The differentially applied external forces do not necessarily have to be
"contact" forces though. They can be just about any kind of
*non*gravitational force desired. For instance they can be forces due to
electric and magnetic fields acting on charges, polarizations,
magnetizations, and currents in and on the sensing body as well the usual
"contact" forces on the body's outer surface. The only kind of gravitational
forces that could conceivably cause a response of such a sensor is that from
a tidal distortion from a *non*uniform gravitational field. In this case the
sensor may have to be enormous in size for such a tidal effect to be
detected. Since a uniform gravitational field will tend to push uniformly on
the sensing medium it cannot contribute (differentially) to the medium's
strain. It is, after all, a kinematic artifact of looking at the sensor
from a noninertial (non-free-falling) coordinate system.

Thus in conclusion, such force/acceleration sensors respond to the internal
(elastic-like) distortion of a sensing medium which responds to the internal
stress which is established as an equilibrating response to differentially
applied external nongravitational forces.

Whether or not such a device can be used as an accelerometer to measure the
component of acceleration along some axis or not depends on whether the
device monitors the inhomogeneity (or at least the anisotropy) of the
resulting distortions. A bathroom scale would make a poor accelerometer
because it only responds to the net compression of its "top" and "bottom"
surfaces. It would effectively give same reading if it was turned upside
down and reused, or if the direction of the acceleration was reversed.
Devices which are supposed to be used as accelerometers can detect the sign
and magnitude of the acceleration along some direction because they can
detect either the anisotropy or the inhomogeneity of the distortion of the
sensing medium along the fiducial axis as well as the magnitude of that
distortion. My own body can make a crude accelerometer. I can tell
that my posterior is subject to a more intense stress (due to the chair's
force) than my head is. I thus conclude that I'm being accelerated up by my
chair.

BTW, since gravitational forces (locally) are not "real" but are artifacts of
the use of a noninertial frame used to describe the physics, I prefer *not*
to define the concept of "weight" as (the magnitude of) the gravitational
force on an object. Such a definition makes an object's weight depend on the
coordinate system of the observer. I prefer to define weight as what others
define as "apparent weight" which is simply the magnitude of the
*non*gravitational force of support on a body which prevents a free fall
state (or equivalently, which deflects the motion from a free fall state).
This definition of weight is independent of the frame of the observer and
agrees with our usual sensations of weight.

David Bowman
dbowman@gtc.georgetown.ky.us