Shell theorem: Difference between revisions

Revision as of 19:50, 13 December 2023

Revision as of 16:39, 26 December 2023

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saying that the gravitational force is the same as that of a point mass in the center of the shell with the same mass.

=== Spherical Shell to Solid Sphere ===

Finally,It integrateis allpossible to use this spherical shell result to re-derive the solid sphere result from earlier. This is done by integrating an infinitesimally thin spherical shell with mass of {{nowrap|<math>dM</math>,}} and we can obtain the total gravity contribution of a solid ball to the object outside the ball

:<math>F_{total} = \int dF_r = \frac{Gm}{r^2} \int dM.</math>

BetweenUniform density means between the radius of <math>x</math> to {{nowrap|<math>x+dx</math>,}} <math>dM</math> can be expressed as a function of {{nowrap|<math>x</math>,}} i.e.,

:<math>dM = \frac{4 \pi x^2 dx}{\frac{4}{3} \pi R^3} M = \frac{3Mx^2 dx}{R^3}</math>

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:<math>F_\text{total} = \frac{3GMm}{r^2 R^3} \int_0^R x^2 \, dx = \frac{GMm}{r^2}</math>

whichAs found earlier, this suggests that the gravity of a solid spherical ball to an exterior object can be simplified as that of a point mass in the center of the ball with the same mass.

== Inside a shell ==