Potential inside homogenous sphere

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To calculate gravitational potential inside a homogeneous sphere of mass m, it is essential to recognize that the gravitational force depends only on the mass closer to the center, as the forces from outer mass cancel out. The sphere's volume is (4/3)πR^3, leading to a density of (3/4)m/(πR^3). The mass below a radius r is given by m(r/R)^3, resulting in a force of -Gm(r/R)^3M/r^2, which simplifies to -Gmr/R^3. Consequently, the gravitational potential energy is expressed as -Gmr^2/(2R^3). This understanding is crucial for generalizing gravitational calculations in non-diametric tunnels through a sphere.
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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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