Classical mechanics ~ Potential energy and periodic movement

[rmfw]

Hey all,

suppose there's a particle with Potential Energy : U(x) = A*[ x^(-2) - x^(-1) ] , where A is a constant.

I'm supposed to find the energy required to make the particle go from periodic movement to unlimited movement.

First thing I did was U '(x) = 0 to find the balance points, now the problem is that there's only one root to the function, which means there's only one balance point at x=2 (stable).

I thought I was going to find two or more balance points to determine the energy that divides the movements.

It's my first time posting here so I hope I'm making myself clear, but can someone explain me where I'm wrong ?

 

[ehild]

Hi rmfw, welcome to PF.

There are some rules here in the Forums, how to post your question. Read https://www.physicsforums.com/showthread.php?t=686781, please.

That particle moves in a potential well, how does it look like? Make a sketch. Why do you think there should be more balance points? What is that equilibrium point, minimum or maximum? At what energy is the particle free to go to infinity and not confined into the well? ehild

 

[rmfw]

Hi, on the first picture is the graph I got. On the second picture is the graph of what I expected. I know that if the particle was trapped in the potential well (picture 2) , with an energy of E>k the particle would go into unlimited movement.

Now back to the first picture, if someone questions me what's the energy required to change the movement of the particle from periodic to unlimited, is the right answer E=0 ?

EDIT: in the second picture both maximums are supposed to be at the same height.

 

[ehild]

Hi, on the first picture is the graph I got. On the second picture is the graph of what I expected. I know that if the particle was trapped in the potential well (picture 2) , with an energy of E>k the particle would go into unlimited movement.

Now back to the first picture, if someone questions me what's the energy required to change the movement of the particle from periodic to unlimited, is the right answer E=0 ?

EDIT: in the second picture both maximums are supposed to be at the same height.

Your first picture is almost right: Assuming A>0, the potential increases to infinity if x -->0.
The potential has got a negative minimum value, and tends to 0 if x -->∞. When the energy of the particle is negative, Emin<E<0, the particle is confined in the potential well. Your answer is correct, if E=0 the movement of the particle becomes unlimited, it can go to positive infinity.

The second picture is not relevant to the problem.

ehild

 

[paranormal]

Hello,

Thank you for sharing your question about potential energy and periodic movement. It seems like you are on the right track with finding the balance points by setting the derivative of the potential energy function to zero. However, in this case, there is only one balance point at x=2 because the potential energy function is a decreasing function for values greater than 2 and an increasing function for values less than 2. This means that there is only one point where the potential energy is at a minimum, and that is at x=2.

To find the energy required to make the particle go from periodic movement to unlimited movement, you can use the conservation of energy principle. The total energy of the system, which includes the kinetic energy and potential energy, will remain constant. So, to make the particle go from periodic movement to unlimited movement, you would simply need to add enough energy to overcome the potential energy barrier at x=2. This added energy would then be converted into kinetic energy, allowing the particle to move without being bound by the potential energy function.

I hope this helps clarify the situation for you. Keep up the good work in your scientific studies!