Problem $4(10$ pts$)$ Refer to Fig. $4$ where a rod $($length $\backslash (\backslash$ell $\backslash ),$ with negligible mass$)$ carries a mass $\backslash ($ m $\backslash )$ at its upper end. It is supported by a linear spring $($spring constant $\backslash ($ k $\backslash ),$ and length $\backslash ($ x$\_\{0\}\backslash )).$ Assume the displacements, $\backslash ($ x $\backslash ),$ and rotations, $\backslash (\backslash$varphi $\backslash ),$ are small. Assume there is gravity $\backslash ($ g $\backslash ).$

$($a$)$ Determine the acceleration of the particle with respect to the inertial frame. Express it in terms of $\backslash (\backslash$varphi $\backslash ).$ Express it in the inertial frame $\backslash ($ E $\backslash ).(2$ pts$)$

$($b$)$ Determine all the forces acting on the mass. Express it in terms of $\backslash (\backslash$varphi $\backslash ).$ Express them in the frame $\backslash ($ E $\backslash ).(2$ pts$)$

$($c$)$ Determine the equations of motion of the particle, where the kinematic variable is $\backslash (\backslash$varphi$($t$)\backslash ).(2$ pts$)$

$($d$)$ Solve the differential equation for $\backslash (\backslash$varphi$($t$)\backslash )$ assuming it is displaced from its vertical position $($small displacement$/$angle$)$ and then released with no initial velocity. $(2$ pts$)$

$($e$)$ What is the resonant frequency of the system $\backslash (\backslash$omega $\backslash )?$ Discuss the three potential cases. $(2$ pts$)$