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Math Help - interesting problem in fluid mechanics

  1. #1
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    Acceleration of sphere

    Here's a rather interesting problem in fluid mechanics.
    A hollow weightless infinitely thin spherical shell (yes, this IS an ideal problem)
    is evacuated and tied with a string to the earth's surface, where the gravitational acceleration is g. Of course, there will be an upward tension in the string of m*g, where m is the mass of the displaced air.
    The string is then cut. Assuming the air is an ideal gas, what is the INITIAL
    upward acceleration of the sphere? Some would say g, but this is wrong,
    since it doesn't account for the acceleration of the air adjacent to the
    sphere.
    The problem was reportedly given to Feynman, and, after working on it for a
    number of hours, came up with the answer. I won't tell you now what he got, but will leave it for those who might want to try to solve it. I don't know how Feynman did it, but will post his answer later on.
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  2. #2
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    Quote Originally Posted by bobbyk
    Here's a rather interesting problem in fluid mechanics.
    A hollow weightless infinitely thin spherical shell (yes, this IS an ideal problem)
    is evacuated and tied with a string to the earth's surface, where the gravitational acceleration is g. Of course, there will be an upward tension in the string of m*g, where m is the mass of the displaced air.
    The string is then cut. Assuming the air is an ideal gas, what is the INITIAL
    upward acceleration of the sphere? Some would say g, but this is wrong,
    since it doesn't account for the acceleration of the air adjacent to the
    sphere.
    I won't answer the main part of this question as I know what the added mass
    for a sphere is under these conditions. But I will discuss the expectation that
    the acceleration would be -g.

    The simple minded analysis would be: In the absence of added mass the
    force acting on sphere is -mg, where m is the mass of displaced air,
    and there is a minus sign because g is downwards acting and the
    buoyancy force is upwards. The mass being accelerated is near zero, so
    the acceleration should be very large, and not -g. (identically zero
    mass as required by the thought experiment would give infinite acceleration).

    The problem was reportedly given to Feynman, and, after working on it for a
    number of hours, came up with the answer. I won't tell you now what he got, but will leave it for those who might want to try to solve it. I don't know how Feynman did it, but will post his answer later on.
    RonL
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  3. #3
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    Sphere acceleration

    You are the only one to have answered my question about the sphere
    acceleration, so I will ask you in a very friendly way, will you post your
    answer, and then I'll tell you what Feynman found.
    Look, this is not a competition --- I think you are one of the greatest
    on this forum!
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  4. #4
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    Sphere acceleration

    OK, since you don't want to post your solution, I'll tell you what Feynman
    found. The initial upward acceleration is 2g.
    My friend's brother was at Caltech and a student of Feynman and he told
    me about this.
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  5. #5
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    Quote Originally Posted by bobbyk View Post
    OK, since you don't want to post your solution, I'll tell you what Feynman
    found. The initial upward acceleration is 2g.
    My friend's brother was at Caltech and a student of Feynman and he told
    me about this.
    Sorry I didn't reply before, I missed the previous post (presumably it occured
    either at a time when I was preoccupied with other things or partiularly
    p***** off with MHF).

    The added mass of a sphere is:

    <br />
M_a=\frac{2}{3}\pi\, r^3\, \rho<br />

    which is half the mass of the displaced air, so the upward accelleration is twice that
    expected of a body of mass equal to the mass of displaced air subject to the same
    boyancy force, or 2g

    RonL

    (however I would like to know how Feynman deduced this, as the added mass
    calculation is not something I can do in my head.)
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  6. #6
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    Sphere acceleration

    Great, Captain, it increases my admiration for you, and would increase it
    even more if you had replyed before I told you what Feynman found!
    (only kidding)
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  7. #7
    Grand Panjandrum
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    Quote Originally Posted by bobbyk View Post
    Great, Captain, it increases my admiration for you, and would increase it
    even more if you had replyed before I told you what Feynman found!
    (only kidding)
    If Feynman's solution did not depend on the form of the cavity
    being a sphere, he will of course have had a fallacious argument.

    In the case of a cylinder the added mass is the mass of the displaced
    air/fluid. in which case the upward acceleration would have been g

    RonL
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  8. #8
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    Sphere acceleration

    Sadly, since my friend, his brother, and Feynman are all dead now, I have no way of finding out how Feynman did it. He was one of the greatest, though,
    as I'm sure you'll agree.
    Last edited by CaptainBlack; December 7th 2006 at 12:01 AM.
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  9. #9
    Grand Panjandrum
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    Quote Originally Posted by bobbyk View Post
    Sadly, since my friend, his brother, and Feynman are all dead now, I have no way of finding out how Feynman did it. He was one of the greatest, though,
    as I'm sure you'll agree.

    Which is why I am interested in how he tackled this. His methods are always illuminating

    RonL
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