(adsbygoogle = window.adsbygoogle || []).push({}); a particle of mass m moves in a potential V(r) = gamma/r^2 where the constant gamma >0

a) Sketch an effective potential and discuss the nature of the non zero angular momentum (L not zero) orbits without solving for the equation of the trajectory

i know that [tex] V_{eff} = V(r) + \frac{L^2}{2mr^2} [/tex]

here [tex] V_{eff} = \frac{1}{r^2} (\gamma + \frac{L^2}{2m}) [/tex]

so it will look like an inverse square graph as in teh digram

Calculate teh equation of the trajectories discuss their shapes and sketch a typical trajectory

one thing thats got me with this question is that i cant solve for r(t) or even r(phi) becuase their formulas involve using E which depends on (dr/dt)^2 and i dont end up getting anything solvable. Have a look

for r(t)

[tex] r(t) = \sqrt{\frac{2}{m} (E - V_{eff}(r))} [/tex]

but [tex] E = \frac{1}{2} m \dot{r}^2 + V(r) + \frac{L^2}{2mr^2} [/tex]

and that yeilds nothing useful

am i doing something wrong? Is energy supposed to be zero? But why?

there are two more parts which i will post later on. they are related to a and b.

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# Homework Help: Particle moving in a potential

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