Understanding SHM in a Potential Well: F(x+xo) and dU/dx Link Explained

AI Thread Summary
The discussion focuses on the relationship between force (F) and potential energy (U) in the context of simple harmonic motion (SHM) within a potential well. It highlights the equation F = -dU/dx and seeks clarification on the derivation that shows the force constant is equivalent to the second derivative of potential energy. The initial differentiation of U = 1/2 kx^2 yields the linear force law, while a second differentiation is necessary to establish the connection to the force constant. Understanding these derivations is crucial for grasping the underlying principles of SHM in potential wells. The conversation emphasizes the mathematical foundation linking force and potential energy in this context.
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In the attachment below some form of link is made between F(x+xo) and dU/dx

I understand F=-dU/dx but I do not understand the derivation shown to prove that the force constant is equal to the second derivative in the last line.

How do they go about this proof ?
ImageUploadedByPhysics Forums1365065923.973605.jpg
 
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As you saw, if you start with U = 1/2 kx^2 and differentiate once, you get the linear force law you want.
Differentiate it a second time.
 

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