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Two particles (with reduced mass $\mu$) that are orbiting each other interact via a potential energy $U = \frac{1}{2}kr^2$, where *k* > 0 and *r* is the distance between them.

(a) Find the equilibrium distance *r*_{0} at which the two particles can circle each other at a constant distance of separation as a function of the angular momentum *L*.

(b) Determine if this is a position of stable equilibrium.

(c) Without solving an orbit equation, determine if those orbits are closed.

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