Question

A mass m moves in a central potential U(r) = −A e^(−kr), where A and k are positive constants.

(a) Sketch the effective potential, and list the types of motion possible (bounded vs.

unbounded, stable vs. unstable) and the energy ranges for each.

(b) If the particle moves in a circular orbit of radius r0, find its angular momentum L in terms

of m, k, A and r0.

(c) Find the angular velocity of the circular orbit ωorb in terms of m, k, A and r0.

(d) Find the frequency of radial oscillations ωosc in terms of k, r0, and ωorb.

(e) By inspection of (d), show that perturbed circular orbits are closed only for certain discrete values of r0. Find the value of r0 that corresponds to an elliptical orbit when the circular orbit is perturbed.

Answer 1

a)   the types of motion possible -   bounded and unstable .

the energy ranges = [−Are^(−kr) , Are^(−kr)]

b) its angular momentum L   = (A * 2pi * r0 * m)/k

c)    the angular velocity of the circular orbit ωorb = (k * m)/(A * 2pi * r0)

d)   the frequency of radial oscillations ωosc =   ωorb/(2pi * r0 * k)

e)   value of r0 that corresponds to an elliptical orbit =   sqrt(2A * m/k)