8.13*** Two particles whose reduced mass is µ interact via a potential energy U = ½kr², where r is the distance between them. (a) Make a sketch showing U (r), the centrifugal potential energy Ucf(r), and the effective potential energy Ueff (r). (Treat the angular momentum l as a known, fixed constant.) (b) Find the "equilibrium" separation ro, the distance at which the two particles can circle each other with constant r. [Hint: This requires that dUeff/dr be zero.] (c) By making a Taylor expansion of Ueff (r) about the equilibrium point ro and neglecting all terms in (r-ro)³ and higher, find the frequency of small oscillations about the circular orbit if the particles are disturbed a little from the separation ro.

8.13*** Two particles whose reduced mass is µ interact via a potential energy U = ½kr², where r is the distance between them. (a) Make a sketch showing U (r), the centrifugal potential energy Ucf(r), and the effective potential energy Ueff (r). (Treat the angular momentum l as a known, fixed constant.) (b) Find the "equilibrium" separation ro, the distance at which the two particles can circle each other with constant r. [Hint: This requires that dUeff/dr be zero.] (c) By making a Taylor expansion of Ueff (r) about the equilibrium point ro and neglecting all terms in (r-ro)³ and higher, find the frequency of small oscillations about the circular orbit if the particles are disturbed a little from the separation ro.