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A particle constrained to move along the x-axis has a potential energy described by U(x) = k[1 - exp(-x²)], where k is a positive constant and -∞ ≤ x ≤ +∞. Which of the following statements is true?

1. The particle is in an unstable equilibrium at positions away from the origin.
2. For any finite, nonzero x, the force on the particle points outward from the origin.
3. If the total mechanical energy is k/2, the particle's kinetic energy is least at the origin.
4. The motion near x = 0 is simple harmonic for small displacements.

Correct answer: The motion near x = 0 is simple harmonic for small displacements.

Solution

U(x)=k[1-exp(-x^2)] has U'(x)=2kx*exp(-x^2), giving a stable minimum at x=0 where U=0. For small x, U approximately k*x^2, so the motion is simple harmonic. The stored claim that KE is least at the origin is false (KE is maximum there since U is minimum).

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