Maximal Gravity

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This Physics quiz website by Yacov Kantor provides the solution in the February 2002 quiz. The optimal surface profile (with max gravity in the origin) in spherical and cylindrical coordinates for the solid of revolution is $r^2=z_0^2 \cos\theta$ and $(z^2+\rho^2)^{3/2}= z_0^2z$, respectively, $0\leq z\leq z_0$. The gravity in the origin is only 2.6% larger than the gravity on the surface of a spherical planet.

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Bernhard Heijstek
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Bernhard Heijstek

Updated on November 01, 2020

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  • Bernhard Heijstek
    Bernhard Heijstek about 3 years

    I found this interesting problem in Introduction to Classical Mechanics with Problems and Solutions by David Morin:

    Given a point $P$ in space, and given a piece of malleable material of constant density, how should you shape and place the material in order to create the largest possible gravitational field at $P$?

    Any ideas?

    • Qmechanic
      Qmechanic over 12 years
      This Physics quiz website by Yacov Kantor provides the solution in the February 2002 quiz. The optimal surface profile (with max gravity in the origin) in spherical and cylindrical coordinates for the solid of revolution is $r^2=z_0^2 \cos\theta$ and $(z^2+\rho^2)^{3/2}= z_0^2z$, respectively, $0\leq z\leq z_0$. The gravity in the origin is only 2.6% larger than the gravity on the surface of a spherical planet.
    • Bernhard Heijstek
      Bernhard Heijstek over 12 years
      @Qmechanic: Thanks! Could you make your comment an answer so that I could accept it?
    • Georg
      Georg over 12 years
      What is a "large" field in this context? BTW, has this curve (or the solid) a special name?
    • JustThinking
      JustThinking about 6 years
      This was also the third problem given in the 2003 Finnish-Estonian Olympiad. You can find the solution here: ioc.ee/~kalda/ipho/E_S.html
  • Ted Bunn
    Ted Bunn over 12 years
    If you want to minimize the potential, rather than maximizing the field strength, the best you can do is to pack the material into a sphere, with the point $P$ at the center. After all, for any other shape, it'd be possible to move a mass element from larger to smaller distance, reducing the potential.
  • Ted Bunn
    Ted Bunn over 12 years
    The most surprising thing to me is how small the improvement is over a simple sphere.
  • Earth is a Spoon
    Earth is a Spoon almost 9 years
    Yes, it does violate constant density condition..
  • Admin
    Admin about 5 years
    Welcome to Physics SE! Please don't post formulas as pictures or plain text, but use MathJax instead. MathJax is easy for people on all devices to read, and can show up clearer on different screen sizes and resolutions. I've edited it here as an example. Look at this Math SE meta post for a quick tutorial.