Potential Energy Homework: Solving F(r) = -k(r-a)

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Homework Help Overview

The problem involves determining the potential energy U(r) from the central force F(r) = -k(r-a), where 'a' represents the natural length of a spring and 'k' is the spring constant. The original poster attempts to integrate the force to find the potential energy but expresses confusion regarding the resulting graph.

Discussion Character

  • Exploratory, Conceptual clarification

Approaches and Questions Raised

  • The original poster integrates the force equation to derive the potential energy but questions the validity of the resulting parabolic shape. Some participants suggest considering the behavior of the potential energy for values of r greater than and less than a.

Discussion Status

Participants are engaging in a constructive dialogue, with some providing clarifications regarding the nature of the potential energy graph. There is an acknowledgment of the original poster's confusion, and responses indicate a shared understanding of the potential energy's characteristics.

Contextual Notes

The discussion includes considerations of the potential energy behavior in relation to the spring's natural length and the implications of the force equation. There is no explicit consensus on the interpretation of the graph, but participants are exploring the implications of the derived function.

oldspice1212
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Homework Statement


The net force on the mass is the central force F(r) = -k(r-a). Find the potential energy U(r).
a is the spring of natural length and k is spring constant k.

Homework Equations


-dU/dx = F(x)

The Attempt at a Solution


F(r) = - \frac{ dU }{ dr } implies -k(r-a) = - \frac{ dU }{ dr }
So I did this and then integrating both sides I got
\frac{ kr^2 }{ 2} - kar = U(r) but this doesn't make much sense when I graph it out, as it gives me a parabola, and I would be expecting a asymptotic type function?
 
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Well think about what happens when r > a and r < a. Both states have potential energy and would therefore be parabolic with the vertex at r = a

P. S. I can't see your pics because I'm on my phone
 
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I didn't upload any pics, was just making sure, and yes that makes sense, thank you :)
 
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oldspice1212 said:
I didn't upload any pics, was just making sure, and yes that makes sense, thank you :)
 
Last edited:
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Yes, that's must easier to see! Thank you so much!
 

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