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Consider a spherical planet of radius R, and let the radial coordinate r originate from the plant's center. If the density of the planet is ρ from 0 ≤ r < R/2 and ρ/3 from R/2 < r < R, then my work tells me that the maximum force due to gravity inside the planet is at r = R/2, not at r = R as one might expect.

[tex]0\leq r\leq R/2,\qquad F_G = \frac{Gm\left(\rho \frac{4}{3}\pi r^3\right)}{r^2} = \frac{4\pi}{3}Gm\rho r \\

R/2 \leq r \leq R,\qquad F_G = \frac{Gm\left[ \rho\frac{4\pi}{3}\left(\frac{R}{2} \right)^3 + \frac{\rho}{3}\frac{4\pi}{3} \left( r^3 - \left(\frac{R}{2} \right)^3 \right) \right]}{r^2} = \frac{4\pi}{3}\frac{Gm\rho}{r^2}\left[ \left(\frac{R}{2} \right)^3 + \frac{1}{3}r^3 - \frac{1}{3}\left(\frac{R}{2} \right)^3 \right][/tex]

My work is above. Is this be correct, that the maximum force due to gravity would be at r = R/2? Thanks for your time!