This is a very basic summary of simple pendulum. I wrote this only for a brief review of LaTeX syntax. For more serious one, refer to this article.
Torsion Pendulum
We hope to find the moving function. i.e. $\theta$ and $t$.
\begin{align*}
\vec{\tau} &= -\kappa \vec{\theta} \
\vec{\tau} &= I \vec{\alpha}
\end{align*}
From definition, we know that:
$$ \ddot{\theta} = \alpha $$
So, we obtain the final function:
$$ \ddot{\theta} = - \frac{\kappa}{I} \theta $$
and the final result:
$$ \theta = \theta_0 \cdot \sin \left( \sqrt{\frac{\kappa}{I}}t + \phi \right) \quad(*1) $$
Pendulum
We define the $R_{CM}$ as the distance from the center of mass to the hanging point, which is only dependent on the shape.
We define $X_{CM}$ the displacement from central axis to the current position (careful for the direction).
$$ \begin{align} \vec{\tau} &= \overrightarrow{R_{CM}} \times \overrightarrow{G} \ &= \overrightarrow{X_{CM}} \times \overrightarrow{G} \end{align} $$
Only consider the case that $\theta$ is small enough, and only consider the magnitude, we have:
$$ X_{CM} = R_{CM} \sin{\theta} \approx R_{CM} \theta $$
And if we consider the direction, for small $\theta$,
$$ \overrightarrow{X_{CM}} \approx - \overrightarrow{R} \times \overrightarrow{\theta} $$
Since
$$ \vec{\tau} = I \vec{\alpha}, $$
Then we get the final formula (magnitude only):
$$ \begin{align} \ddot{\theta} &\approx - \frac{R_{CM} \cdot Mg}{I} \theta \ \Rightarrow \theta &= \theta_{0} \sin \left( \sqrt{\frac{R_{CM} \cdot Mg}{I}} t , + \phi \right) \qquad (*2) \end{align} $$