Question

A point mass moves without friction on the inside of a surface of revolution z=f(r) whose axis of symmetry is along a gravitational field 
-g

a)Find the condition for a stationary circular orbit of radius $r_{0}$ and show that It is stable or unstable under small impulses along a transverse surface
to the direction of movement according to whether it is positive or negative.
For a stable orbit, find the frequency ω of small oscillations around the balance setting.
Original Question(in spanish)

Answer 1

along gravitational field of the Giren surface of mevolution z f(ro); and symmetry quis is Gar) so potential enengy system stem V (0) x f (0) let constant angularo momentum L, mass, m effective potential energy, repp =V (or) + now for a stationary oro bit, then 2mp2 aveff of radius ro ar Ip=r ,2 mro3 2 >> | f'(r.) mno condition for stationary orbit Now, for stable orbit, 22 Veff 70 f"(r.) дүр 2- 312 m Tч 199 ino >o d (putt value of L mr.3 (stable onbit) or, no f" (ro) + 3 f'(2) >0 (stable on for unstable equilibrium alleff Әр 2 =r. po f" (70) + 3 f Cro) <0 (un stable frequency of small as cillation for stable bit is, W = Parvepre ro-ro lm on bit) )}/r Im = {"(r.) + 3P%C4