Chapter 8: Problem 13
Two particles whose reduced mass is \(\mu\) interact via a potential energy \(U=\frac{1}{2} k r^{2},\) where \(r\) is the distance between them. (a) Make a sketch showing \(U(r),\) the centrifugal potential energy \(U_{\mathrm{cf}}(r)\) and the effective potential energy \(U_{\text {eff }}(r) .\) (Treat the angular momentum \(\ell\) as a known, fixed constant.) (b) Find the "equilibrium" separation \(r_{\mathrm{o}},\) the distance at which the two particles can circle each other with constant \(r .\left[\text { Hint: This requires that } d U_{\text {eff }} / d r \text { be zero. }\right]\left(\text { c) By making a Taylor expansion of } U_{\text {eff }}(r)\right.\) about the equilibrium point \(r_{\mathrm{o}}\) and neglecting all terms in \(\left(r-r_{\mathrm{o}}\right)^{3}\) and higher, find the frequency of small oscillations about the circular orbit if the particles are disturbed a little from the separation \(r_{\mathrm{o}}.\)
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