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-   -   Using geometry to avoid calculus (https://www.mersenneforum.org/showthread.php?t=9283)

davieddy 2007-09-15 14:34

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.

davieddy 2007-09-15 15:49

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

cheesehead 2007-09-16 10:50

[quote=davieddy;114317]Show that a spherical shell of mass M attracts an external mass m as would a point mass M at its centre.[/quote]Newton's [i]Principia[/i] includes a proof, of course. :)

davieddy 2007-09-16 11:11

[quote=cheesehead;114377]Newton's [I]Principia[/I] includes a proof, of course. :)[/quote]

Yes. But "avoiding calculus" wasn't one of his claims to fame:smile:

davieddy 2007-09-16 11:42

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.

cheesehead 2007-09-18 08:38

[quote=davieddy;114379]Yes. But "avoiding calculus" wasn't one of his claims to fame:smile:[/quote]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 [I]Principia[/I]. I guess my "of course" was misleading.

davieddy 2007-09-18 10:00

[quote=cheesehead;114545]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 [I]Principia[/I]. I guess my "of course" was misleading.[/quote]

Did he use my trick?


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