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How far will sphere fall into a paraboloid? (uses curvature I think )

  1. Oct 17, 2009 #1
    1. The problem statement, all variables and given/known data
    A sphere with a radius of 4 is dropped into a paraboloid. How far is the bottom point of the sphere from the bottom point of the paraboloid when the sphere stops falling? What is the radius of the largest sphere that will fall all of the way and touch the bottom of the paraboloid (distance=0)?


    2. Relevant equations
    Paraboloid (fixed location with nubby bit touching origin)
    z=x2+y2

    Sphere (falls with gravity until it gets stuck, center unknown)
    x2+y2+z2=42=16

    3. The attempt at a solution
    The was an optional "challenge" problem and the solutions from the back of the book are (to two decimal places) 12.25 and 0.50 respectively. This comes from the chapter about curvature so I'm pretty sure it involves comparing the curvature of the "container" paraboloid with the sphere that you "drop in". Since the sphere has no fixed center it seems to me that you need to compare the curvatures. My intuition is that there is a circle around the paraboloid that will be a "pinch point" and stop the sphere from falling anymore. At such a point the paraboloid and sphere would have the same curvature...I think. I can't figure out how to take it any farther than that.
     
  2. jcsd
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