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  • Buoyancy Problem II

    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
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