Nonlinear spring $(10$ points$)$

For a mass$-$spring system, suppose we have a non$-$linear spring that exerts a force described by the

following relation:

$F_s=0\mathrm{.}5-\frac{1}{1+e^{-5x}}$

$($a$)$ Plot the force for the range of displacements $x=[-1,1]$

$($b$)$ Find the EOM for this spring$-$mass system.

This EOM is a little more difficult to solve. But if our model represents a system that does

not move beyond the range of $x=[-0\mathrm{.}2,0\mathrm{.}2]$ we can "linearize" our spring force and assume a

relationship $F_s=-kx.$

$($c$)$ What value would you choose for $k$ for this linearized model?

$($d$)$ Find the EOM using the linearized relationship.