View Full Version : Center of buoyancy problem

01-25-1995, 12:18 PM
Dear Colleagues:

I have a dilemma that I hope someone out there can help me with.
Scott McLean and I are trying to measure the center of buoyancy of a
human body at the surface of the water (with application to floating
and swimming). First we measure the center of mass (CM)location for
our subjects using a reaction board. Then we place them in the pool
in a either a prone or a supine floating position. For most people,
their feet tend to sink in this position, indicating that the buoyant
force (B) is less than body weight (W) and that the center of
buoyancy (CB) is cranial to the CM. So we place a strap around their
ankles and measure how much force (R) is required to keep the feet
up. A free body diagram is shown below (as best I can draw it using
7-bit ascii characters):

CB \|/------x-------|
Head ___________._____.___________________ Feet
^ CM ^
| |
|--d--| |

In the above FBD, we know W, R, and x. We would like to know d (the
distance between the CM and CB). Using equations of static
equilibrium for translation (B=W-R) and rotation (Bd=Rx), we can
arrive at B and then d (no big deal).

However, if we move the point of application of the supporting force
(R) higher up the leg, keeping the body position the same, we
decrease the distance x. Everything else about the body SHOULD have
stayed the same, i.e., the same body position in the water, same body
weight, same buoyant force, and same locations of the CM and CB. To
satisfy the rotational equilibrium condition, Bd=Rx regardless of the
value of x. So if x gets smaller then R must increase to produce the
same torque. However, [AND THIS IS WHERE I AM HAVING PROBLEMS], if R
gets larger, then the only way to satisfy the translational
equilibrium condition (B=W-R) is for B to decrease. This doesn't
make sense, however, because the buoyant force (B) is (by definition)
the weight of the displaced water which has not changed. If W, B,
and the locations of CB and CM have not changed, then we have an
impossible situation. We cannot satisfy both the rotational and
translational equilibrium equations. CAN ANYONE OUT THERE SEE THE

NOTE: We have tried this with the body totally submerged and on the
surface and for different amounts of air in the lungs. We have also
put supports at two locations (with a strap around the chest as well
as the ankles) and have come up with the same dilemma; the FBD is
only slightly more complicated).

Any help you can provide will be appreciated. Thanks in advance. I
will post a summary of responses.


Richard N. Hinrichs, Ph.D.
Dept. of Exercise Science
Arizona State University USA
(1) 602-965-1624 (office)
(1) 602-955-8108 (fax)
Hinrichs@asu.edu (email)