Problem in central force motion

[Vashist Settipalli]

Homework Statement

a particle moves in a circle under the influence of an inverse cube law force. Show that the particle can also move with uniform radial velocity,either in or out.
Find theta as a function of r for motion with uniform radial velocity.[/B]

Homework Equations

f=-2A/r^3[/B]
ueff=l^2/2mr^2

The Attempt at a Solution

i found the effective potential but could not reach any conclusion though through that.
the solution states that if the effective potential becomes 0,the radial motion becomes uniform.
i don't get this
please help.

 

[Orodruin]

You did not find the effective potential. The expression

ueff=l^2/2mr^2

only contains the angular momentum part of the effective potential. What happened to the potential of the inverse cube law force?

If the effective potential in the radial coordinate is a constant, then the equation of motion for the radial coordinate will become $m\ddot r = - U_{\rm eff}'(r) = 0$.

 

[Vashist Settipalli]

You did not find the effective potential. The expression

only contains the angular momentum part of the effective potential. What happened to the potential of the inverse cube law force?

If the effective potential in the radial coordinate is a constant, then the equation of motion for the radial coordinate will become $m\ddot r = - U_{\rm eff}'(r) = 0$.

the equations i can take care of,hopefully.
I don't get why effective potential defines the motion since there is already a force in the radial direction.
TIA

 

[Orodruin]

Sorry, but it is not clear what you want to say. The force you are referring to is the radial force in three dimensions. You can rewrite this problem as an effective problem in one dimension by integrating out the angular motion. This introduces the angular momentum barrier into the effective potential for the one-dimensional radial motion, which therefore has a potential that relates both to the inverse cube law force and to the angular momentum barrier.

 

[Vashist Settipalli]

Sorry, but it is not clear what you want to say. The force you are referring to is the radial force in three dimensions. You can rewrite this problem as an effective problem in one dimension by integrating out the angular motion. This introduces the angular momentum barrier into the effective potential for the one-dimensional radial motion, which therefore has a potential that relates both to the inverse cube law force and to the angular momentum barrier.

Thank you very much.
I forgot that we were trying to change mtion in 3d to 1d and kept pondering over the fact that if the radial motion is uniform,then the given 3d force should become 0(silly me).
Anyways thank you very much.
you made my day