Potential inside homogenous sphere

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SUMMARY

The gravitational potential inside a homogeneous sphere of mass m can be calculated using the formula for potential energy, which is -Gmr²/(2R³). This derivation relies on the understanding that the gravitational force within the sphere is proportional to the distance from the center, r, and that mass outside this radius does not contribute to the gravitational force. The volume of the sphere is (4/3)πR³, and the density is (3/4)m/(πR³). This analysis is crucial for solving problems related to gravitational fields in non-diametric tunnels through celestial bodies.

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llandau
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How to calculate gravitational potential inside a homogenous sphere of mass m? I am curious because I had to solve the classic problem of the tunnel through Earth and wanted to generalize when the tunnel is not a diameter.
 
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The gravitational force inside a homogeneous sphere depends only on the mass closer to the center- the force vectors due to mass farther out cancel each other.

If the radius of the sphere is R, then its volume is (4/3)\pi R^3 and the density is (3/4)m/(\pi R^3). That means that the mass of the sphere below radius r is (3/4)m/(\pi R^3)(4/3)\pi r^3= m(r/R)^3. The force on an object of mass M then is -Gm(r/R)^3M/r^2= -Gmr/R^3.

So the force is proportional to r and the potential energy is -Gmr^2/(2R^3).
 
You have been helpful, thanks (I believe you lost an M, but it really doesn't matter).
 

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