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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Author by
Bernhard Heijstek
Updated on November 01, 2020Comments
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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?
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Qmechanic over 12 yearsThis 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 over 12 years@Qmechanic: Thanks! Could you make your comment an answer so that I could accept it?
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Georg over 12 yearsWhat is a "large" field in this context? BTW, has this curve (or the solid) a special name?
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JustThinking about 6 yearsThis 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
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Ted Bunn over 12 yearsIf 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.
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Ted Bunn over 12 yearsThe most surprising thing to me is how small the improvement is over a simple sphere.
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Earth is a Spoon almost 9 yearsYes, it does violate constant density condition..
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