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A particle oscillates about the equilibrium position x 0 subject to a force that has an associated potential energy U(x). Which of the following statements (perhaps more than one ) about U(x) is true? (A) U(x) must be symmetric about x 0 . (B) U(x) must have a minimum at x 0 . (C) U(x) may have a maximum at x 0 . (D) U(x) must be positive in the vicinity of x 0 . A particle oscillates about the equilibrium position x0 subject to a force that has an associated potential energy U(x). Which of the following statements (perhaps more than one ) about U(x) is true?(A) U(x) must be symmetric about x0.(B) U(x) must have a minimum at x0.(C) U(x) may have a maximum at x0.(D) U(x) must be positive in the vicinity of x0.
The correct option is (A), (B) and (D).Since the particle oscillates about the equilibrium position x0, therefore the potential energy U(x) must be symmetric about x0. Thus option (A) is correct.The force and potential energy is related by Fx(x) = -dU/dx. The force must always act to restore the particle to equilibrium, and the potential energy must have a minimum at the equilibrium position. Thus option (B) is correct.Since the particle oscillates about the equilibrium position x0, so the potential energy U(x) must be symmetric about x0 and the potential energy must have a minimum value at the equilibrium position. Therefore U(x) must be positive in the vicinity of x0. Thus option (D) is correct.
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