Problem $2(20$ points$)$: $($Similar to Problem B$-3-19$ from K$.$ Ogata's System Dynamics,

$4$th Edition but not the same$).$

Referring to the spring$-$loaded inverted pendulum

system in Figure $2,$ obtain the natural frequency ${}_n$

of the system using an energy method. Choose the

potential energy at the equilibrium state to be zero.

Notice that the rod of the pendulum is NOT

MASSLESS but it has a homogeneously distributed

mass $m_R.$

Fig. $2.$ Spring$-$loaded inverted

pendulum. The mass of the rod

of the pendulum is $m_R.$