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Designing Planets (Posted on 2019-01-04) Difficulty: 4 of 5
You are asked to design a planet of mass M and uniform density p such that it has the maximum possible acceleration due to gravity g at a point on its surface.

Find the shape of this planet, and then find g.

Take G as the universal gravitational constant.

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 1.0000 (1 votes)

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more research | Comment 2 of 5 |
I was not able to solve this myself. While I recalled that the "calculus of variations", e.g., the use of Lagrange Multipliers, is often used for such optimizations, I had forgotten how they worked. That is given here

A very friendly solution to this nice problem is here

Be sure to see the second article about outreach and The Little Prince.
The second article also references a purely calculus-based solution.  

But, I wish I has done it myself! 

Edited on January 10, 2019, 1:28 am
  Posted by Steven Lord on 2019-01-04 21:46:05

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