mersenneforum.org Using geometry to avoid calculus
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 2007-09-15, 14:34 #1 davieddy     "Lucan" Dec 2006 England 647410 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 2007-09-15 at 15:13
 2007-09-15, 15:49 #2 davieddy     "Lucan" Dec 2006 England 2×3×13×83 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 2007-09-15 at 16:12 Reason: Too trivial to mention
2007-09-16, 10:50   #3

"Richard B. Woods"
Aug 2002
Wisconsin USA

1E0C16 Posts

Quote:
 Originally Posted by davieddy Show that a spherical shell of mass M attracts an external mass m as would a point mass M at its centre.
Newton's Principia includes a proof, of course. :)

2007-09-16, 11:11   #4
davieddy

"Lucan"
Dec 2006
England

2×3×13×83 Posts

Quote:
 Originally Posted by cheesehead Newton's Principia includes a proof, of course. :)
Yes. But "avoiding calculus" wasn't one of his claims to fame

 2007-09-16, 11:42 #5 davieddy     "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.
2007-09-18, 08:38   #6

"Richard B. Woods"
Aug 2002
Wisconsin USA

22×3×641 Posts

Quote:
 Originally Posted by davieddy Yes. But "avoiding calculus" wasn't one of his claims to fame
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 2007-09-18 at 08:42

2007-09-18, 10:00   #7
davieddy

"Lucan"
Dec 2006
England

194A16 Posts

Quote:
 Originally Posted by cheesehead 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.
Did he use my trick?

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