The differential equation that gives the equation of motion of a pendulum where:
- $m$ is the mass
- $L$ is the distance between the pivot and the body's centre of mass
- $g$ is the acceleration due to gravity
- $I$ is the moment of inertia of the body about the pivot
is given by:
$\displaystyle\frac{\partial^2 \theta}{\partial t^2} + \left(\frac{mgL}{I}\right) \sin\left(\theta\right) = 0$
Here, we're going to neglect air resistance and friction.
I plugged this equation into Wolfram Alpha and the solution seems to be:
$\displaystyle\theta = 2\,\text{am}\left(\frac{t + \omega_0}{2} \sqrt{\theta_0 + \frac{2mgL}{I}} \,\,\bigg| \,\,\frac{4mgL}{2mgL + I\theta_0}\right)$
where $\text{am}\left(x, y\right)$ is the Jacobi amplitude function.
But, plugging the numbers, the units don't cancel out and they aren't in the right order. Now, my question becomes: Is this the correct equation of motion? If not, what is it? Do units not matter when plugging into the equation of motion?