A particle moves in 1-dimension with potential energy given by U(x)=Uo(1−e−b2x2) where Uo and b are positive constants.
a) Find the force associated with this potential energy F(x).
b) Plot F(x) and U(x) on the same axes (you will need to give numerical values to Uo and b).
c) The particle is released from rest from the point x=1/b. Using conservation of energy, find the velocity of the particle when it passes through x=0 in terms of the given constants.
d) For very small displacements around x=0, show that the potential energy expression describes simple harmonic motion, and then find the period of small oscillations about x=0 in terms of the given constants (you will need to use a Taylor series expansion to do this).