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Keep in mind the following things: Newton's Law. Newton's Law Hooke's Law.

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Writing it the way you did, you don't try to compensate the force, but to set it equal. This means you account for the fact that the centripetal force is gravity and nothing more and they do point in the same direction for a circular orbit. Assume a body in circular motion around the origin of a coordinate system at distance $R$. One can express this motion ...

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Hints: Consider the potential $$\tag{1} V_{\rm cf}(r)~:=~\frac{\hbar^2}{2m} \frac{\ell(\ell+1)}{r^2}.$$ For which radius $r$ is the potential (1) smallest? If it smallest for large $r$, it is reasonable to call it a centrifugal (rather than a centripetal) potential. Alternatively, what is the sign of the corresponding force \tag{2} F_{\rm ...

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