Interpreting a Potential Energy Graph

In summary, the potential energy graph shows that a particle moving in one dimension is subject to the potential energy curve shown in the plot. If the particle has total energy of 3 Joules at point xc and its velocity is in the positive x direction, it will eventually reach point xa. If the particle has kinetic energy of 3 Joules at point xc it can reach point xb. If the total energy of the particle at point xb is 3 Joules it can reach point xc.
  • #1
moxy
41
0
[SOLVED] Interpreting a Potential Energy Graph

Homework Statement



A particle moving in one dimension is subject to the potential energy curve shown in
the plot. Which of the following statements is true?

[Diagram attached]

A) At point xa, the force on the particle is in the negative x direction.
B) If the particle has total energy of 3 Joules at point xc and its velocity is in the positive
x direction, it will eventually reach point xa.
C) If the particle has kinetic energy of 3 Joules at point xc it can reach point xb.
D) If the total energy of the particle at point xb is 3 Joules it can reach point xc.
E) If the particle is released from rest at xc, it will travel in the positive x direction.

Homework Equations



F = -dU/dx

The Attempt at a Solution



I know A is false because F is the negative derivative of U. The slope at point Xa is negative, so when I take the negative of that, F ends up being positive.

Other than that, the wording confuses me. I'm sure that it's a whole lot simpler than I'm making it out to be.

I'm not just looking for the answer (I know it's C), I'd like to understand why the other choices are false.
 

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  • #2
Hi Moxy

I think you could interpret the graph a lot simpler. As the potential energy is plottet agains the direction x, you can simply think about the whole thing as kind of a "hill" (or several "hills" in this case). If you look at the statements, it should work:

A) If you would release the particle at x_a, it would roll down the 'hill' in the positive x direction -> the force is in the positive direction. (Your approach with F=-dU/dx works perfectly here too)

B)The potential Energy of the particle at the point x_c is 1 Joule, with the given total energy of 3 Joules the kinetic Energy has to be 2 Joules (assuming there's no friction). As the graph goes higher than 3 Joules between x_b and x_c, the particle cannot reach x_b or x_a.

C)Potential energy + kinetic energy = 4 Joules -> particle can reach x_b

D)+E) should be clear with this...

I don't know if this helps... Sorry for my poor English
 
  • #3
B) is false. At point xc, the potential energy measures up to 1J, and if the total energy measures up to 3J, then 2J goes to kinetic energy. Due to the conservation of energy, the particle cannot reach point xa; between points xc and xb, there is a hill that measures beyond 3J...to get past that point, it needs at least that amount of energy, which it does not.

D) is false for the same reason B is false; there is a hill between points xb and xc, in which more than 3J is needed.

E) is false because F = -dU/dx, in which the force there will point in the negative x-direction (and from rest, the particle has no initial velocity).

C) is the correct choice because at point xc, there is 1J of potential energy and 3J of kinetic energy. The total energy measures up to 4J, which is more than the amount needed to surpass the hill between xc and xb.
 
Last edited:
  • #4
Okay, those both make sense. I kind of figured the "peak" in the graph between Xb and Xc had something to do with it, but I wasn't sure exactly what it implied. I figured the question was a lot simpler than I was making it. The hill analogy made it much easier to conceptualize.

Thanks!
 

1. What is a potential energy graph?

A potential energy graph is a visual representation of the relationship between potential energy and distance or position. It shows how the potential energy of a system changes as the distance between objects or the position of a single object changes.

2. What does the shape of a potential energy graph indicate?

The shape of a potential energy graph indicates the type of potential energy present in a system. A steep slope indicates a strong force, while a shallow slope indicates a weak force. A flat slope indicates no force or a constant force.

3. How is potential energy calculated from a graph?

To calculate potential energy from a graph, you need to identify the relevant points on the graph and use the equation PE = mgh (for gravitational potential energy) or PE = ½kx² (for elastic potential energy) where m is mass, g is acceleration due to gravity, h is height, k is the spring constant, and x is the displacement.

4. What factors can affect the shape of a potential energy graph?

The shape of a potential energy graph can be affected by the distance or position of objects, the type of potential energy (gravitational, elastic, etc.), the forces acting on the system, and any changes in mass or spring constants.

5. Why is it important to understand potential energy graphs?

Potential energy graphs are important in understanding the behavior of systems and predicting the outcomes of interactions between objects. They are essential in many fields of science, including physics, chemistry, and biology, and can help us understand and model complex systems in the natural world.

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