A) At point away from the origin, the particle is in unstable equilibrium
B) For any finite non-zero value of x, there is a force directed away from the origin
C) If its total mechanical energy is k/2, it has its minimum kinetic energy at the origin
D) For small displacements from x = 0, the motion is simple harmonic
Correct Answer: D
Solution :
Potential energy of the particle \[U=k(1-{{e}^{-{{x}^{2}}}})\] Force on particle\[F=\frac{-dU}{dx}=-k[-{{e}^{-{{x}^{2}}}}\times (-2x)]\] F\[=\,-2kx{{e}^{-{{x}^{2}}}}\]\[=-2kx\left[ 1-{{x}^{2}}+\frac{{{x}^{4}}}{2\,!}-...... \right]\] For small displacement \[F=-2kx\] Þ \[F(x)\propto -x\] i.e. motion is simple harmonic motion.
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