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When a particle of mass m moves on the x-axis in a potential of the form V(x) = kx2, it performs simple harmonic motion. The corresponding time period is proportional to √(m/k), as can be seen easily-using dimensional analysis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of x = 0 in a way different from kx2 and its total energy is such that the particle does not escape to infinity. Consider, a particle of mass m moving on the x-axis. Its potential energy is V(x) = ax4 (α > 0) for |x| near the origin and becomes a constant equal to V0 for |x| ≥ X0 (see figure).If the total energy of the particle is E, it will perform periodic motion only if(a) E < 0 (b) E > 0 (c) V0 > E > 0 (d) E > V0 |
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Answer» Correct Answer is: (c) V0 > E > 0 For an oscillating system. The kinetic energy must be zero periodically, for a finite value of x. Also, E < V0 . |
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