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A particle free to move along the x-axis has potential energy given by U(x) = k for -∞≤x≤+∞, where k is a positive constant of appropriate dimensions. Then?

A particle free to move along the x-axis has potential energy given by U(x) = k for -∞≤x≤+∞, where k is a positive constant of appropriate dimensions. Then? Correct Answer 1-exp -x2

U=k(1-e(-x2)) F=-dU/dx=-2kxe(-x2)=-2kx(1-x2+⋯) For small x, F ≅-2kx This shows that the force is directed towards the origin and for smaller x, F∝x. Hence the motion is simple harmonic.

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