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==2022-05-12==

$$\begin{cases}
I \ddot \theta = 0 = k_1 z (\hat z \wedge \hat L) \frac{L}{4} - k_2 z (\hat z \wedge \hat L) \frac{3L}{4} \\
mg\hat z = -(k_1 + k_2) z \hat z
\end{cases}$$

$$\begin{cases}
I \ddot \theta = 0 = \frac{1}{4} zL (k_1 - 3 k_2) (\hat z \wedge \hat L)\\
mg\hat z = -(k_1 + k_2) z \hat z
\end{cases}$$