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

[oldspice1212]

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?

 

[Brian T]

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

 

[oldspice1212]

I didn't upload any pics, was just making sure, and yes that makes sense, thank you :)

 

[Brian T]

I didn't upload any pics, was just making sure, and yes that makes sense, thank you :)

 

[oldspice1212]

Yes, that's must easier to see! Thank you so much!