Shell theorem: Difference between revisions

Revision as of 04:47, 29 November 2020

Revision as of 04:48, 29 November 2020

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(Note: the <math>d\theta</math> in the diagram refers to the small angle, not the [[arclength|arc length]]. The arc length is <math display="inline">R\ d\theta</math>.)

Applying [[Newton's Universal Law of Gravitation]], the sum of the forces due to the mass elements in the shaded band is<blockquote><math>dF = \frac{Gm \;dM}{s^2}</math></blockquote>However, since there is partial cancellation due to the [[Euclidean vector|vector]] nature of the force in conjunction with the circular band's symmetry, the leftover [[Vector (geometry)#Vector components|component]] (in the direction pointing towards <math>m</math>) is given by<blockquote><math>dF_r = \frac{Gm \;dM}{s^2} \cos\phivarphi</math></blockquote>The total force on <math>m</math>, then, is simply the sum of the force exerted by all the bands. By shrinking the width of each band, and increasing the number of bands, the sum becomes an integral expression:

{{block indent|<math>F_r = \int dF_r</math>}}

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Since <math>G</math> and <math>m</math> are constants, they may be taken out of the integral:

{{block indent|<math>F_r = Gm \int \frac{\cos\phivarphi\ dM} {s^2}. </math>}}

To evaluate this integral, one must first express <math>dM</math> as a function of <math>d\theta</math>

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and

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

By the [[law of cosines]],

{{block indent|<math>\cos\phivarphi = \frac{r^2 + s^2 - R^2}{2rs}</math>}}

and

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{{block indent|<math>\cos\theta = \frac{r^2 + R^2 - s^2}{2rR}.</math>}}

These two relations link the three parameters <math>\theta</math>, <math>\phivarphi</math> and <math>s</math> that appear in the integral together. As <math>\theta</math> increases from <math>0</math> to <math>\pi</math> radians, <math>\phivarphi</math> varies from the initial value 0 to a maximal value before finally returning to zero at&nbsp;<math>\theta=\pi</math> . At the same time, <math>s</math> increases from the initial value <math>r-R</math> to the final value <math>r+R</math> as <math>\theta</math> increases from 0 to <math>\pi</math> radians. This is illustrated in the following animation:

[[File:Shell-diag-1-anim.gif|center]]

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It follows that

{{block indent|<math>F_r = \frac{GMm}{2} \frac{1}{rR} \int \frac{s\cos\phivarphi} {s^2}\,ds = \frac{GMm}{2rR} \int \frac{\cos\phivarphi} {s}\,ds </math>}}

where the new integration variable <math>s</math> increases from <math>r-R</math> to&nbsp;<math>r+R</math> .

Inserting the expression for <math>\cos \phivarphi</math> using the first of the "cosine law" expressions above, one finally gets that

{{block indent|<math>F_r = \frac{GMm}{4r^2 R} \int \left( 1 + \frac{r^2 - R^2}{s^2} \right)\ ds\ .</math>}}

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so:

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

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