A Force and Potential-Energy Problem

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A conservative force being zero with respect to position x does not imply that potential energy U must also be zero. It is possible to have potential energy without an active force acting towards equilibrium, as demonstrated by a ball resting at the top of a hill. In this scenario, the force is zero, but the potential energy is at a maximum. The relationship between force and potential energy indicates that a zero force results in no change in potential energy for small displacements. Thus, potential energy can exist independently of the presence of a force.
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Homework Statement



My book says that just because a conservative force is zero with respect to x, doesn't necessarily mean that the potential-energy U has to be zero.

How can it be possible to have potential energy without a force acting towards the equilibrium?

Homework Equations



Fx(x) = dU(x)/dx = 0

The Attempt at a Solution



Fx(x)*Δx = -U, so 0*Δx = 0
 
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As your DEs show, it only implies that the change in PE is vanishingly small for small changes in x. Consider e.g. a ball sitting on the top of a hill.
 

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