This exercise illustrates how a seemingly simple system can exhibit a rich array of dynamic behavior. Consider the system shown below. The slider of mass $m$ slides on the rail without friction. It is attached to the ground by a linear spring with a spring constant $k.$ The unstretched length of the spring is $\sqrt[\phantom{2}]{2}l.$ Use the skeleton code on eLC to answer the following questions about this system.

Questions:

Optional Extra Credit, $10$ points: Show that the equation of motion for this system is

$m{x}^{}+(1-\frac{\sqrt[\phantom{2}]{2}l}{\sqrt[\phantom{2}]{l^2+x^2}})kx=0.$

Notice that the equation of motion of this system is nonlinear. What part of the equation of motion tells you that it is nonlinear? Explain in words why you think it is possible for this system $-$ which only has linear spring $-$ to have a nonlinear equation of motion.

By inspecting the diagram of this system, you should be able to see that it has three equilibrium positions. In terms of the degree of freedom $x,$ what are these positions? Note that an equilibrium position in this system is one where the spring force in the $x-$direction is zero.

Optional Extra Credit, $10$ points: Let $l=1$ and set the potential energy datum $($where $V=0)$ at $x=1.$ Show that the potential energy function for the system is

$V=\frac{x^2}{2}-\sqrt[\phantom{2}]{2(x^2+1)}+\frac{3}{2}$