A Force and Potential-Energy Problem

[student34]

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

 

[haruspex]

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.