- #1
stunner5000pt
- 1,461
- 2
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 that's got me with this question is that i can't solve for r(t) or even r(phi) becuase their formulas involve using E which depends on (dr/dt)^2 and i don't 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.
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 that's got me with this question is that i can't solve for r(t) or even r(phi) becuase their formulas involve using E which depends on (dr/dt)^2 and i don't 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.