Pendulum $($thin rod$)$

$($you may check Problem A$-3-8,$ which is for a pointmass, not a thin rod$).$ Draw the FBD and write the nonlinear EOM of a pendulum consisting of a thin, $4-kg$ stick of length $L$ suspended from a pivot at a point $\text{'}O\text{'}.$ If $$ is small enough to assume that $sin$~$=,$ how long should the rod be in order for the period to be exactly $2$ sec $?$

$($You will find a linearized EOM of the form: ${}^{}+2{}_n{}^{}+{}_n^2=0.$ The oscillation period for this $2^{nd}$ order system is: $T-=2\frac{}{{}_n})$

The moment of inertia for a thin circular rod of length $L$ and mass $m$ is:

$I_C=\frac{m{L}^2}{12}$ about the center of mass $\text{'}$ C $\text{'}{I}_0=\frac{m{L}^2}{3}$ about an end point $\text{'}O\text{'}$

Figure $1.$ Mass$-$Spring$-$Damper Systems

Foucault Pendulum in the Pantheon in Paris