20070915, 14:34  #1 
"Lucan"
Dec 2006
England
6474_{10} Posts 
Using geometry to avoid calculus
Show that a spherical shell of mass M attracts an external mass m as would a point mass M at its centre.
The simplest argument uses Gauss' theorem about flux, and symmetry. To show it directly we sum the force exerted by each element of area in the shell on m. How would you do this? David PS assume Newton's law of gravitation. Last fiddled with by davieddy on 20070915 at 15:13 
20070915, 15:49  #2 
"Lucan"
Dec 2006
England
14512_{8} Posts 
I realize it is a challenge to talk about geometrical problems without diagams, but most of us are up to this.
I can't remember whether it was Laplace, Lagrange or Legendre who prided himself on producing a treatise containing no diagrams. Perhaps his name didn't begin with L or wasn't even French:) David PS Any enlightenment gratefully received Last fiddled with by davieddy on 20070915 at 16:12 Reason: Too trivial to mention 
20070916, 10:50  #3 
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}·3·641 Posts 

20070916, 11:11  #4 
"Lucan"
Dec 2006
England
6474_{10} Posts 

20070916, 11:42  #5 
"Lucan"
Dec 2006
England
2·3·13·83 Posts 
Let R be the radius of the shell, and r be the distance
of m from the centre of the shell. Let P be a point displaced from the centre towards m by a distance R^2/r. Now express the contribution to the resultant force made by an element of the shell in terms of the solid angle it subtends at P. 
20070918, 08:38  #6 
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}×3×641 Posts 
I meant that that proof _is_ by geometry, not calculus. Newton didn't run around proving everything by means of his new calculus; geometric proofs are common in Principia. I guess my "of course" was misleading.
Last fiddled with by cheesehead on 20070918 at 08:42 
20070918, 10:00  #7 
"Lucan"
Dec 2006
England
6474_{10} Posts 

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