Determine the equivalent parameter using $\theta$ co-ordinate, when the system is given counter clockwise displacement from SEP [ system equilibrium position] was generalized co-ordinate.

$\rightarrow$ $KE = (KE)_rod + (KE)_Bob$

$\frac{1}{2} (Io) \theta^2 + \frac{1}{2} (Io) \theta^2$

$= \frac{1}{2} [I_G + m_{rod} (dist)^2] \theta^2 + \frac{1}{2} (mpL2) \theta^2$

$= \frac{1}{2} [\frac{m_L^2}{12} + \frac{m_L^2}{4}] \theta^2 + \frac{1}{2} mpL^2 \theta^2$

$= \frac{1}{2} [ \frac{m_L^2}{3} + mpL^2] \theta^2$

$= \frac{1}{2} [ + mpL^2] \theta^2$

$= \frac{1}{2} [ \frac{mL^2}{3} + mpL^2] \theta^2$

$PE = (PE)_k + (PE)_bpb + (PE)_rod$

$= \frac{1}{2} k (\frac{2L\theta}{3})^2 + mpgL (1- cos \theta) + mg \frac{1}{2} (1- cos \theta)$

We know that

$cos \theta \approx 1 - \frac{\theta^2}{2}$

$PE = \frac{1}{2} k [\frac{2L}{3} \theta]^2 + mg \frac{L}{2} [ 1 - ( 1 - \frac{\theta^2}{2})] + mp.g.L [ 1 - ( 1 - \frac{\theta^2}{2})]$

$= \frac{1}{2} k. \frac{4L^2 \theta^2}{9} + mg \frac{L}{2} + mp.g.l [ 1 - 1 + \frac{\theta^2}{2}$

$= \frac{1}{2} [ \frac{k.4 L^2 \theta^2}{9} + \frac{mgL \theta^2}{2} + mp.g. L \theta^2$

$= \frac{1}{2} [ 4kL^2 + \frac{mgL}{2} + m.p.g.L] \theta^2$

$K_eq = 4kL^2 + \frac{mgL}{2} + mp.g.L$