Expression for effective potential energy

[Thorscira]

<< Mentor Note -- Poster has been reminded to use the Template when starting new schoolwork threads >>

Two particles of identical mass m interact with each other via central potential energy

Vcentral(r) = -V0(1-|r|/a), if 0 <= |r| <= a
0, if a < |r|

Constants are V0 > 0 and a > 0

What's the expression for the effective potential energy and what are the constants in your expression?

My attempt:

Veff = -V0(1-|r|/a) + M/2mr^2

M is the moment of inertia which is constant/conserved in any system relative to the centre.

/*I'm not sure about this any and all help would be very much appreciated! Thank you in advance :) */

 

[BvU]

Hello Thorscira $\qquad$ $\qquad$ !

Please don't erase the template -- guidelines

I don't agree with

M is the moment of inertia which is constant/conserved in any system relative to the centre.

there is something else (involving M) that is conserved

The expression (Please use the sub- and superscript buttons)
V_eff = -V₀(1-|r|/a) + M/2mr²
can not be right: check the dimensions !

 

[Thorscira]

The expression (Please use the sub- and superscript buttons)

I'm so sorry! It's my first time posting on here and I wasn't sure how to define vectors and things. r is a vector describing two dimensions.

 

[BvU]

can you fix the dimensions problem I mentioned ?

 

[Thorscira]

V_effective(r) = -V₀(1-|r|/α) + M/2mr²

I hope this is okay, I underlined the vectors.

 

[BvU]

second term is length squared

 

[Thorscira]

This is what I've got so far:

V_effective(r) = -V₀(1-|r|/α) + M/2μr²

In this case the angular momentum would be constant and μ, right?

If we keep the distance α fixed but V₀ can be varied and the angular momentum is nonzero. Is there a way one can express the critical value of V₀ as a function of the reduced mass, α, and the angular momentum?

 

[BvU]

you can not add length squared to energy
is it clear what mu is?