Question

# A particle free to move along the x-axis has potential energy given by $$U(x)=k[1-exp\left ( -x^{2} \right )]$$ for$$-\infty \leq x\leq +\infty ,$$ 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.

Solution

## The correct option is D for small displacements from x = 0, the motion is SHM.$$exp \left ( -x^{2} \right ) = 1 - x^{2} + \cfrac{x^{4}}{2!} + ---$$For  small  $$x$$ :  $$exp \left ( -x^{2} \right ) = (1 - x^{2})$$Thus  $$U(x) = K [1-(1-x^{2})] = K x^{2}$$     $$F = - \dfrac{dU}{dx} = - \dfrac{d(kx^{2})}{dx} = -2 K x$$Thus, the motion  is  an  SHM.Physics

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