Chapter 4.P2, Problem 2P

(a) Find the linearization of $\text{Eq}\text{.}\left(\text{i}\right)$ at $x=0$.

(b) In the case that $h>0$, what is the natural frequency of the linearized system ? Explain how the natural frequency of the linearized system depends on $h$ and $L$.

(c) On the same set of coordinate axes, plot the graphs of

$y=f\left(x\right)=2kx\left[1-\frac{L}{\sqrt{{\left(L+h\right)}^2+x^2}}\right]$,

its first degree Taylor polynomial $y_1=f^{\prime }\left(0\right)x$, and its third degree Taylor polynomial $y_3=f^{\prime }\left(0\right)x+f^{″}\left(0\right)x^2/2+f^{‴}\left(0\right)x^3/6$.

Construct these plots for each of the following sets of parameter values:

i.     $L=1,k=1,h=1$

ii.   $L=1,k=1,h=0.5$

iii.  $L=1,k=1,h=0.1$

iv.   $L=1,k=1,h=0$

(d) Find the dependence on $h$ of the approximate range of values of $x$ such that $\left|y-y_1\right|\approx \left|y_3-y_1\right|<0.1$.

(e) In the case that $h=0$, explain why the solution of the linearized equation obtained in Problem $2$ is a poor approximation to the solution of the nonlinear equation $\left(\text{i}\right)$ for any initial conditions other than $x\left(0\right)=0,x^{\prime }\left(0\right)=0$.

$y=f\left(x\right)=2kx\left[1-\frac{L}{\sqrt{{\left(L+h\right)}^2+x^2}}\right]$ (i)