A particle free to move along the $x-$axis has potential energy given by $U(x) = k[1 - \exp {( - x)^2}]$ for $ - \infty \le x \le + \infty $, where k is a positive constant of appropriate dimensions. Then
IIT 1999,AIIMS 1995, Diffcult
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(d)Potential energy of the particle $U = k(1 - {e^{ - {x^2}}})$
Force on particle$F = \frac{{ - dU}}{{dx}} = - k[ - {e^{ - {x^2}}} \times ( - 2x)]$
F$ = \, - 2kx{e^{ - {x^2}}}$$ = - 2kx\left[ {1 - {x^2} + \frac{{{x^4}}}{{2\,!}} - ......} \right]$
For small displacement $F = - 2kx$
$⇒$ $F(x) \propto - x$ i.e. motion is simple harmonic motion.
Force on particle$F = \frac{{ - dU}}{{dx}} = - k[ - {e^{ - {x^2}}} \times ( - 2x)]$
F$ = \, - 2kx{e^{ - {x^2}}}$$ = - 2kx\left[ {1 - {x^2} + \frac{{{x^4}}}{{2\,!}} - ......} \right]$
For small displacement $F = - 2kx$
$⇒$ $F(x) \propto - x$ i.e. motion is simple harmonic motion.
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