Question

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

• A. at points 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 SHM.

Answer 1

The correct option is D for small displacements from x = 0, the motion is SHM.

$exp(-x^2)=1-x^2+\frac{x^4}{2!}+---$
For small $x$ : $exp(-x^2)=(1-x^2)$
Thus $U\left(x\right)=K[1-(1-x^2\left)\right]=K{x}^2$
$F=-\frac{dU}{dx}=-\frac{d\left(k{x}^2\right)}{dx}=-2Kx$
Thus, the motion is an SHM.