Question

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

• A. for small displacement from $x=0$, the motion is simple harmonic
• B. if its total mechanical energy is $k/2$, it has its minimum kinetic energy at the origin.
• C. for any finite nor-zero value of $x$, there is a force directed away from the origin.
• D. at points away from the origin, the particles is in unstable equilibrium.

Answer 1

Answer: (A)

Since $F=-\frac{dU}{dx}=2kx\exp \left(-x^2\right)$
$F=0$ (at equilibrium as $x=0$ )
$U$ is minimum at $x=0$ and $U_{\min }=0$
$U$ is maximum at $x\to \pm \infty$ and $U_{\max }=k$
The particle would oscillate about $x=0$ for small displacement from the origin and it is in stable equilibrium at the origin.