Shell theorem: Difference between revisions

Revision as of 16:36, 14 March 2021

Revision as of 16:41, 14 March 2021

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In fact, this allows exactly one more class of force than the (Newtonian) inverse square.<ref name=Gurzadyan>{{cite journal| last=Gurzadyan |first=Vahe |authorlink=vahe Gurzadyan|title=The cosmological constant in McCrea-Milne cosmological scheme|journal=The Observatory|date= 1985|volume=105|pages=42–43|bibcode=1985Obs...105...42G}} http://adsabs.harvard.edu/full/1985Obs...105...42G</ref><ref name=Arens>{{cite journal| last=Arens| first=Richard| authorlink=Richard Friederich Arens|title=Newton's observations about the field of a uniform thin spherical shell|journal=Note di Matematica|date=January 1, 1990|volume=X|issue=Suppl. n. 1|pages=39–45}}</ref> The most general force as derived in <ref name="Gurzadyan"/> is:

{{block indent|:<math> F = -\frac{G M m}{r^2} - \frac{\Lambda M m r}{3} </math>}}

where <math>G</math> and <math>\Lambda</math> can be constants taking any value. The first term is the familiar law of universal gravitation; the second is an additional force, analogous to the [[cosmological constant]] term in [[general relativity]].

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Another generalization can be made for a disc by observing that

{{block indent|:<math>dM=\frac{R^2}{2} \frac{d\theta \sin^2\theta}{\pi R^2}M=\frac{ \sin^2\theta}{2 \pi}M \, d\theta </math>}}

so:

{{block indent|:<math>F_r = \frac{GMm}{2 \pi} \int \frac{ \sin^2 \theta \cos\varphi} {s^2} \, d\theta,</math>}}

where <math>M=\pi R^2 \rho</math>, and <math>\rho</math> is the density of the body.

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Doing all the intermediate calculations we get:

{{block indent|:<math>F(r) = \frac{Gm \rho}{8r^3} \int_{R-r}^{R+r} { \frac{\left(r^2 + s^2 - R^2\right)\sqrt{2\left(r^2R2 R^2 + r^2s2 s^2 + R^2s2 s^2\right) - s^4 - r^4 - R^4} }{s^2} } \, ds</math>}}

== Newton's proofs ==