$1.$ Harmonic Frequency

$-($a$)$ Assuming small oscillations, derive an expression for the harmonic frequency of this system when $\backslash ($ Q $\backslash )$ is close to zero.

$2.$ Comparison of Terms in the Potential

$-($a$)$ Find an expression for the value of $\backslash ($ Q $\backslash )$ at which the fourth$-$order term $\backslash (\backslash$frac$\{1\}\{4\}$ D Q$^\{4\}\backslash )$ becomes larger than the second$-$order term $\backslash (\backslash$frac$\{1\}\{2\}$ K Q$^\{2\}\backslash ).$ Discuss what this indicates about the relative importance of each term at larger amplitudes.

$3.$ Equation of Motion

$-($a$)$ Write down the equation of motion for this system.

$4.$ Amplitude$-$Dependent Frequency $($Computational$)$

$-($a$)$ Numerically solve for the angular frequency of the oscillator as a function of initial amplitude $\backslash ($ Q$\_\{0\}\backslash )($for zero initial velocity$).$ One potential approach is to solve the equation of motion using a differential equation solver and then apply a root$-$finding algorithm to locate the period.

$($b$)$ Plot the frequency as a function of the starting amplitude $\backslash ($ Q$\_\{0\}\backslash ).$

$-($c$)$ Describe and explain how the frequency changes with amplitude.