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Deformation of a spinning sphere

  1. Oct 15, 2012 #1
    1. The problem statement, all variables and given/known dataHello guys! Now this is not a homework question, but it may sound like one. If a uniform sized and massed sphere was spinning in space away from any source of forces that could affect it, wouldn't the only forces that act on it are the centripetal and centrifugal? And if this sphere had elasticity, wouldn't the force that is stretching the sphere be either centripetal of centrifugal force? Now I am not including its gravitational force on purpose. Its gravitational field can be negligible. I also know that the centrifugal force is not a real force, but it is useful for explaining some phenomena. I'm using classical mechanics. The sphere will not be spinning at relativistic speeds.




    2. Relevant equations

    mωv=-kx

    3. The attempt at a solutionWell, mathematically, I figured that if the centripetal and centrifugal forces are the sources of deformation of stretching of the sphere, then by using Hooke's Law we find that
    mωv=-kx
    where mωv is the centripetal or centrifugal force magnitude and -kx is Hooke's Law. Is this right? Or am I forgetting something? I would really appreciate anyone's help. My extremely inferior mind is not sure of itself.:frown:
     
  2. jcsd
  3. Oct 17, 2012 #2
    Yes, this problem can be solved, but not the way you suggested. This is a complicated problem in Theory of Elasticity. You need to use Hooke's Law in its full tensorial form to get the stresses that develop in the sphere. This includes hoop stresses that are important in supporting the centrifugal acceleration. The stresses that develop have to combine properly with the centrifugal forces. The boundary conditions involve the stresses at the surface of the sphere: the stress tensor dotted with a unit normal to the surface must be equal to zero. This is a 2D problem for the latitudinal and radial displacements; the longitudinal displacement is equal to zero. It's an interesting problem, but takes some work to solve.
     
  4. Oct 18, 2012 #3
    Oh I know I could use tensors but I was just wondering ( which I didn't state ) if the equation would be a rough estimate of the deformation on the sphere in the direction of x if x was taken to be on the radius of the sphere. Would it be a very rough estimate?
     
  5. Oct 18, 2012 #4
    If x was taken to be on the same axis as the radius that is. I'm not sure if that in its self is correct though. haha:smile:
     
  6. Oct 18, 2012 #5
    Well the problem would be a rough estimate if instead of a sphere we use a spinning disk, right?
     
  7. Oct 18, 2012 #6
    I will just use the tensor equations instead. Algebra is too limited lol:smile: Thank you very much for your time! You were very helpful!:smile:
     
  8. Oct 19, 2012 #7
    A spinning disk would be a more manageable problem to get a Theory of Elasticity solution for. In the spinning disk problem, you have a state of Plane Stress(which simplifies things), and, in addition, there is only one dependent variable, namely, the radial displacement u. The circumferential hoop strain is u/r, and the radial strain is du/dr. The principal directions of stress and strain are in the radial and hoop directions. So you have an ordinary differential equation to solve for the radial displacement, rather than partial differential equations for two displacements. In the disk problem, the boundary condition at the edge of the disk is zero radial stress.
     
  9. Oct 19, 2012 #8
    Thank you very much! I appreciate your help very much! Thank you for providing insight on this situation:smile:
     
  10. Oct 21, 2012 #9
    Here's an even simpler problem you can try, and it's statically determinate (so you don't have to use Hooke's law). Instead of a disk, consider a ring of circular cross section rotating about its axis. The elements of the ring are obviously accelerating radially, but what is causing the acceleration? There is no radial spring in this picture. In this case, the hoop stress is what is solely responsible for applying the radial acceleration. Do a force balance on a small curved section of the ring (free body), and use Newton's second law to determine the hoop tension.

    Chet
     
  11. Oct 21, 2012 #10
    I never thought about it like that!:smile: Thank you very much! That actually simplified my problem to a great extent! If we decided to use a sphere, would we be able to use the bulk modulus to determine the force stretching the deforming the sphere? And then we can equate that to the equation to the force equation of centrifugal force?:smile:
     
  12. Oct 21, 2012 #11
    No, you couldn't do it that way. The disk problem and the sphere problem are not statically determinate problems. In both these cases, you would need to use the tensorial form of Hooke's law involving stresses and strains, and expressed in terms of the Young's modulus and the Poisson ratio. In the disk problem there would be one displacement to solve for, namely, the radial displacement. In the sphere problem, there would be two displacements to solve for, the radial displacement and the latitudinal displacement. The spherical problem would involve 2 partial differential equations involving the two displacements. I have trouble believing that neither of these problems has been solved in the open literature. I suggest that, if you want to avoid the work of solving the sphere problem, you search the literature (which you should be doing anyway).
     
  13. Oct 21, 2012 #12
    Thank you very much:smile:
     
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