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

Moderate

A

for small displacement from x= 0, the motion is simple harmonic
B

if its total mechanical energy is $\frac{k}{2}$ , 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.

Solution

Since $F = - \frac{dU}{dx} = 2 kx \exp (- x^{2})$
F = 0 (at equilibrium as x = 0)
$U is minimum at x = 0 and U_{\min} = 0$
${U is maximum at x →±∞ and U}_{m a x} = k$
The particle would oscillate about x = 0 for small displacement from the origin and it is in stable equilibrium at the origin.

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