Problem $\backslash$# $4$

Consider the pendulum$-$spring system shown below:

a$)$ Derive the equation of motion and find the natural frequency of the system.

b$)$ Let the mass of the pendulum is $2$ kg $,$ the length of the pendulum is $0\mathrm{.}5$ m and the stiffness of the spring is $\backslash (20\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$m$\}\backslash ).$ Assume that the rod of the pendulum has no mass. The initial angular displacement is $0\mathrm{.}175$ rad $($approx$.10$ degrees$)$ measured from the vertical and there is no initial angular velocity. Find the approximate value of the damping coefficient required for the oscillations to be less than $\backslash (2\backslash \%\backslash )$ of their initial value, within $3$ seconds.