Question

A particle free to move along x axis has potential energy given as $U\left(x\right)=k(1-e^{(-x^2)})$ for $-a\le x\le +\infty$ 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 simple harmonic

Answer 1

The correct option is D For small displacements from x=0, the motion is simple harmonic

The graph of U(x) with x is as shown in figure Potential energy is zero at $x=0$ and maximum at $x=\pm a$
As mechanical energy has fixed value i.e. k/2, the kinetic energy has to be maximum at $x=0$ and maximum at $x=\pm a$.