How can potential energy functions be used to model a system's motion?

In summary: If you have a bead at rest at one end of the wire, and you try to move it to the other end, the bead will follow the wire's potential energy gradient. If you have more than one bead, they will all follow the same direction--the bead on the wire with the lowest potential energy. If you have a system of beads, each with its own potential energy, and you try to move them all to the same point, they will all move in different directions, according to their individual potential energies.
  • #1
LovePhys
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Homework Statement



attachment.php?attachmentid=56061&stc=1&d=1361704966.png


Homework Equations


dU/dx=-Fx
Conservation of energy

The Attempt at a Solution


Here are my solutions. Please feel free to correct me. Any help will be appreciated.
(a) I think that the particle is moving in the negative direction because the potential energy decreases (?)
(b) Use conservation of energy (total energy=4J, from the graph at x=0.5m): min potential energy=1J => max kinetic energy=3J => max speed=12.2m/s
(c)x=2.5m
(d)kinetic energy=4-3=1J => speed=7.1m/s
(e)I think the particle will change direction at the points where the derivative is 0. (I'm thinking about gravitational potential energy).
(f)Again, I think the answer is at points where the derivative is 0 since dU/dx=-Fx

By the way, I think this graph confuses me a bit. I tried without success to imagine the situation, and the teacher told me that it was going to be complicated since the graph was related to molecular bonds. Also, the problem did not state whether it involves only conservative force or not. Can anyone please make a comment on this graph? Thank you very much!

Love Phys
 

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  • #2
(a) If potential energy decreases in the positive x direction, what can be said about the force acting on the particle? What can be said about the motion of the particle under this force if it is at rest initially?

(e) To change direction, the particle must momentarily have zero velocity. Where would that happen?

(f) So where are the points where the derivative is zero?
 
  • #3
Thank you, voko!

(a) I think the force is acting in the positive direction? Therefore, the particle will move in the positive direction.
(b) Is it at x=4.5m? I think that's where potantial energy=4J, so kinetic energy must be 0J.
(c) I reckon it's at x=2.5, and x=5.25?
 
  • #4
All correct.

Is there anything still unclear?
 
  • #5
For (a), the way I came up with the answer is I took the derivative dU/dx<0, therefore Fx=-dU/dx must be positive. But if I think back of how gravitational potential energy is defined in high school, I am confused again. Assume that I have 2 points A and B on the y-axis (y(A)<y(B)) (the positive direction is up), then if I move a particle from B to A, then the gravitational potential energy decreases. And because of the way I choose the positive direction, the gravitational force is acting in the negative direction. So, the particle will move in the negative direction (!?) Sorry this was the kind of picture I had in mind when I answered this question, and obviously it was wrong. Can you please help me understand this? Thank you very much!
 
  • #6
If you sketch this in the same way, the value of the gravitational potential energy on the vertical axis, and the y distance on the horizontal axis, what will the graph of the gravitational potential energy look like?
 
  • #7
I think it's going to be a straight line which passes through the origin with gradient = mg. Then if I want to move from B to A, then I'm doing negative work, but the gravitational force is doing positive work (this is true since vector Fg and vector y are in the same direction).
 
  • #8
The essential feature is that the gradient is positive for gravitation as you go up; in the original problem, however, the gradient in the vicinity of 0.5 m is negative, so the effect of the force is opposite in direction.
 
  • #9
Thank you very much, voko! I think I understand it now. Well, sometimes it's easier to follow the Math logic, instead of trying to imagine what's going on.
 
  • #10
I think it could be helpful to keep in mind that a system always tends to go where the potential energy is lower. If you have a plot of potential energy, imagine it is made of wire with a little bead riding on it. Then the bead's motion can be used to model the system.
 

1. What is a potential energy function?

A potential energy function is a mathematical representation of the potential energy of a system as a function of its position or configuration. It is often used in physics and engineering to model the behavior of systems and predict their energy states.

2. How is a potential energy function related to force?

A potential energy function is related to force through the gradient of the function. The negative of the gradient of the potential energy function with respect to position gives the force acting on the system. This relationship is known as the force-potential energy relationship.

3. What are some common types of potential energy functions?

Some common types of potential energy functions include gravitational potential energy, electrical potential energy, and elastic potential energy. These functions depend on various factors such as distance, charge, and deformation, respectively.

4. How is potential energy function different from kinetic energy?

Potential energy is the energy that a system possesses due to its position or configuration, while kinetic energy is the energy that a system possesses due to its motion. Potential energy can be converted into kinetic energy and vice versa, but they are different forms of energy.

5. What is the significance of the potential energy function in physics?

The potential energy function is significant in physics because it allows us to predict the behavior and stability of systems. It is also an essential concept in the law of conservation of energy, as potential energy can be converted into other forms of energy and vice versa.

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