Quick potential energy questions

[Anza Power]

Given the potential energy between two objects that is function of the distance between the two objects: u=a/r³ - b/r² (a,b>0)

I need to find where the object is in a resting state, and the energy require to completely separate the two objects...

The attempt at a solution
I went ahead and drew a graph for it:

I marked two points in the graph, B is the point when there is no restoring force correct? (as in if you put the object in that point it will not move)

The energy required to separate the two objects completely is U(∞)-u(r) correct?

So if the object was in a point before point A, all you need to do is let go and they'll blast off to infinity? (as in you don't need to put in work to separate them)

Another question (not in the exercise) if the graph looked like this:

For you to separate the two objects does that mean that you only need to get it a fraction beyond point c and they'll blast away from each other?

 

[Redbelly98]

Given the potential energy between two objects that is function of the distance between the two objects: u=a/r³ - b/r² (a,b>0)

I need to find where the object is in a resting state, and the energy require to completely separate the two objects...

The attempt at a solution
I went ahead and drew a graph for it:

I marked two points in the graph, B is the point when there is no restoring force correct? (as in if you put the object in that point it will not move)

Yes.

The energy required to separate the two objects completely is U(∞)-u(r) correct?

Only if they begin with an energy U(r), but it's not clear from what you have said that they do.
The objects must have total energy (kinetic + potential) of U(∞) or greater in order to become completely separated.
Or, you could say that if the objects start with total energy E initially, then an energy of U(∞)-E must be added in order to completely separate the objects.

So if the object was in a point before point A, all you need to do is let go and they'll blast off to infinity? (as in you don't need to put in work to separate them)

Yes.

Another question (not in the exercise) if the graph looked like this:

View attachment 30132

For you to separate the two objects does that mean that you only need to get it a fraction beyond point c and they'll blast away from each other?

Yes, pretty much. As long as they have an energy of at least U(c), they can be separated. And if they are released from rest at point a, they will become separated.

Hope that helps.

 

[Anza Power]

^ That helped a lot, thanks...