Greetings:
Dr. Hinrichs and I were pleased by the response to our previous
posting concerning the center of buoyancy. We now have a problem which may
be more interesting. The method of locating the center of buoyancy was
validated using a 180 cm long piece of PVC pipe of approximately 2 inch
diameter. One end was filled with concrete. The ends were sealed so the
other end was filled with air. The CM of the pipe was measured to be 65 cm
from the heavy end. The CV was at 90 cm or half way along the length of the
pipe due to the symmetry of the pipe.
The measurement of the CB was validated by supporting this pipe with
two straps (one on each side of the CB) while the pipe was fully submerged.
The fully submerged CV would be coincident with the CB. A 4 kg mass was
added to the strap on the light end and a 2 kg mass was added on the heavy
end to keep the pipe submerged. The free body diagram is pictured below.
dB
/\B
/\ R1  /\ R2
dR1  
heavy  CM   light
================================================ ===========
end  CB end
dW


\/W
dR2
where
R1 = supporting force at the heavy end
R2 = supporting force at the light end
B = buoyant force
W = weight
dR1 = distance from heavy end to R1
dR2 = distance from heavy end to R2
dB = distance from heavy end to CB
dW = distance from heavy end to CM
The translational and rotational equations of statis equilibrium for this
system are
sum of forces = 0
eqn (1) R1 + B + R2  W = 0.
sum of moments = 0
eqn (2) (R1*dR1) + (B*dB) + (R2*dR2)  (W*dW) = 0.
Solving these equations for the location of the CB in terms of measured
forces and distances gives
eqn (3) (W*dW)  (R1*dR1)  (R2*dR2)
dB = .
W  R1  R2
The net forces (i.e., with the added weight due to the added masses
subtracted out), R1 and R2 were measured using calibrated load cells. R2
was found to be negative (implying that the application of a downward force
was required to keep this end of the pipe submerged).
The measurement of dB was approximately 90 cm (within 3 or 4 mm) when R1 and
R2 were positioned symmetrically about the CB regardless of the distance
between the load cells. However, when the heavy end support remained
stationary and the light end support was moved towards the middle, the
calculated dB grew linearly from 89.5 cm to 109.8 cm (over 9 locations).
The opposite was found when the light end supporting force remained
stationary and the heavy end supporting force was moved. dB decreased
linearly from 89.5 cm to 69.1 cm over 9 locations.
We suspected that there were errors in the measurements and perhaps caused
the strange results. But upon inspection, the buoyant force was predicted to
be approxmately 3380 g (* 9.81/1000 N). This calculation held for every
condition. This compared well with the calculated volume of the pipe
(3309 cm^3). The systematic (linear) nature of the deviation suggests that
the results were not due to poor data.
The system contains two unknowns, the magnitude of the buoyant force (B) and
the point of application of the buoyant force (dB). If the data we have
collected are real then moving the load cells asymmetrically changes either
the magnitude of the buoyant force (B) or the point of application of this
force. The fact that the measured buoyant force agrees so well with the
theoretical buoyant force would indicate that dB changes. This makes no
sense based on the definitions of the buoyant force and the CB.
Can anyone see an error in the derivations we have made or the logic we are
using? Any help would be greatly appreciated. We will post a summary of
all responses we receive.
Thanks again,
Scott

Scott Mclean
smclean@iastate.edu
Dr. Hinrichs and I were pleased by the response to our previous
posting concerning the center of buoyancy. We now have a problem which may
be more interesting. The method of locating the center of buoyancy was
validated using a 180 cm long piece of PVC pipe of approximately 2 inch
diameter. One end was filled with concrete. The ends were sealed so the
other end was filled with air. The CM of the pipe was measured to be 65 cm
from the heavy end. The CV was at 90 cm or half way along the length of the
pipe due to the symmetry of the pipe.
The measurement of the CB was validated by supporting this pipe with
two straps (one on each side of the CB) while the pipe was fully submerged.
The fully submerged CV would be coincident with the CB. A 4 kg mass was
added to the strap on the light end and a 2 kg mass was added on the heavy
end to keep the pipe submerged. The free body diagram is pictured below.
dB
/\B
/\ R1  /\ R2
dR1  
heavy  CM   light
================================================ ===========
end  CB end
dW


\/W
dR2
where
R1 = supporting force at the heavy end
R2 = supporting force at the light end
B = buoyant force
W = weight
dR1 = distance from heavy end to R1
dR2 = distance from heavy end to R2
dB = distance from heavy end to CB
dW = distance from heavy end to CM
The translational and rotational equations of statis equilibrium for this
system are
sum of forces = 0
eqn (1) R1 + B + R2  W = 0.
sum of moments = 0
eqn (2) (R1*dR1) + (B*dB) + (R2*dR2)  (W*dW) = 0.
Solving these equations for the location of the CB in terms of measured
forces and distances gives
eqn (3) (W*dW)  (R1*dR1)  (R2*dR2)
dB = .
W  R1  R2
The net forces (i.e., with the added weight due to the added masses
subtracted out), R1 and R2 were measured using calibrated load cells. R2
was found to be negative (implying that the application of a downward force
was required to keep this end of the pipe submerged).
The measurement of dB was approximately 90 cm (within 3 or 4 mm) when R1 and
R2 were positioned symmetrically about the CB regardless of the distance
between the load cells. However, when the heavy end support remained
stationary and the light end support was moved towards the middle, the
calculated dB grew linearly from 89.5 cm to 109.8 cm (over 9 locations).
The opposite was found when the light end supporting force remained
stationary and the heavy end supporting force was moved. dB decreased
linearly from 89.5 cm to 69.1 cm over 9 locations.
We suspected that there were errors in the measurements and perhaps caused
the strange results. But upon inspection, the buoyant force was predicted to
be approxmately 3380 g (* 9.81/1000 N). This calculation held for every
condition. This compared well with the calculated volume of the pipe
(3309 cm^3). The systematic (linear) nature of the deviation suggests that
the results were not due to poor data.
The system contains two unknowns, the magnitude of the buoyant force (B) and
the point of application of the buoyant force (dB). If the data we have
collected are real then moving the load cells asymmetrically changes either
the magnitude of the buoyant force (B) or the point of application of this
force. The fact that the measured buoyant force agrees so well with the
theoretical buoyant force would indicate that dB changes. This makes no
sense based on the definitions of the buoyant force and the CB.
Can anyone see an error in the derivations we have made or the logic we are
using? Any help would be greatly appreciated. We will post a summary of
all responses we receive.
Thanks again,
Scott

Scott Mclean
smclean@iastate.edu