Using geometry to avoid calculus

[davieddy]

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]

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]

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. :)

[davieddy]

Originally Posted by cheesehead

Newton's Principia includes a proof, of course. :)

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

[davieddy]

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]

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.

[davieddy]

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?