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James Kenyon Sprague
02-06-1995, 03:57 AM
sorry for the delayed response, I got behind in my mail. I have done a
fair bit of work regarding locating the CM in people and may be able to
explain your problem. Your FBD and equations are correct, but your
assumption that the body orientation and therefore buoyancy force
magnitude (B) and location (d) remain unchanged as the R force moves is
erroneous. My vast experience in unintentionally paddling over
semi-submerged logs and rocks has taught me that a small variation in
externally applied force does cause a canoe to change orientation
untill a geometry is achieved which once again brings the system into
static equilibrium. A similar thing happens with your subjects.

I can visualize it best with a FBD of a dumbell consisting of a
weightless shaft, a hollow sphere of mass m/2 on the left and a small
solid sphere of mass m/2 on the left. When placed in water, the solid
sphere will try to sink, much like a subject's feet. A vertical force (R)
can be applied to the solid sphere via a cord to keep the entire assembly
even (horizontal). You can solve for the buoyancy force and magnitude by
summing forces and moments about the CM: B=Rx/d d=Rx/(mg-R) (as you
already did). The equations are correct, and you can see that R is a
linear function of both R and x, and that d is linear in x and non-linear
in R, so the magnitude and location of the buoyancy force changes with
magnitude and location of the cord force.

The physical interpretation is that any time the cord force is changed,
the body seeks a new position and orientation which produces a buoyancy
force which will satisfy the equilibrium equations... So, even though the
body looks like it is in the same configuration when you move the cord
(change x), in actuality, the system (body or dumbell) changes the wetted
surface, and correspondingly, the submerged volume. You should actually
be able to see this on a person if the water is still.

I might suggest that you perform these experiments with the subject
strapped into a neutral buoyancy "back-board" which remains entirely
submerged for the duration of the testing. This would allow you to
better constrain the subject to your desired configuration with a minimum
of unintended joint motion, and it would give you a good standard
location to which you could fasten your cord, either beneath the heel, or
at the knees, etc...

Good luck,

Jim Sprague, Ph.D. (Mechanical Engineering)