User:Tohline/DarkMatter/UniformSphere
Force Exerted by a Uniform-Density Shell or Sphere
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Tohline's Derivations Circa 1983
If the force per unit mass exerted at the position, <math>~\vec{r}</math>, from a single point mass, <math>~m</math>, is given by,
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<math>~\vec{F}</math> |
<math>~=</math> |
<math>~- \biggl( \frac{G^'m}{r} \biggr) \frac{\vec{r}}{r} \, ,</math> |
then the force per unit mass exerted at <math>~\vec{x}</math> by a continuous mass distribution, whose mass density is defined by the function <math>~\rho(\vec{x}^')</math>, is,
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<math>~\vec{F}(\vec{x})</math> |
<math>~=</math> |
<math>~- \int G^' \rho(\vec{x}^') \biggl[ \frac{\vec{x}^' - \vec{x}}{| \vec{x}^' - \vec{x} |^2} \biggr] d^3x^' \, .</math> |
This central force can also be expressed in terms of the gradient of a scalar potential, <math>~\Phi(\vec{x})</math>, specifically,
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<math>~\vec{F}(\vec{x})</math> |
<math>~=</math> |
<math>~- \vec\nabla\Phi(\vec{x}) \, ,</math> |
where,
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<math>~\Phi(\vec{x}) </math> |
<math>~=</math> |
<math>~ \int G^' \rho(\vec{x}^') \ln | \vec{x}^' - \vec{x} | d^3x^' \, .</math> |
See Also
- Stabilizing a Cold Disk with a 1/r Force Law
- Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?
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© 2014 - 2021 by Joel E. Tohline |