A particle free to move along the x-axis has potential energy given by $U (x) = k [1 - \exp (- x^{2})]$ for $- \infty \leq x \leq + \infty$ , where k is a positive constant of appropriate dimensions. Then

Moderate

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

Solution

Potential energy of the particle $U = k (1 - e^{- x^{2}})$
Force on particle $F = \frac{- dU}{dx} = - k [- e^{- x^{2}} \times (- 2 x)]$
$F = - 2 {kxe}^{- x^{2}} = - 2 kx [1 - x^{2} + \frac{x^{4}}{2!} - . .....]$
For small displacement $F = - 2 kx$
$\Rightarrow F (x) \propto - x$ i.e. motion is simple harmonic motion.

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