I have a question behind the conceptual understanding of the following equation: $$\frac{\text{d}}{\text{d}t}\mathbf{L}_G = \sum_i \ \mathbf{r}_i\times \mathbf{f}_i$$ where $\mathbf{L}_G$ is the total angular momentum of a rigid body about the center of mass $G$, $\mathbf{r}_i$ is the position vector of a point part of the rigid body taken from $G$ and $\mathbf{f}_i$ is the external force at that point.
Suppose we have a cylinder rolling down that hill as shown on pages 86-87 here. Why is it the case that $C\ddot{\theta} = aF$ as shown on page 87? How does one leverage the formula above to arrive at this equation? Conceptually speaking, one sums the moment of the forces about the center of mass for each point over the entire body, so what exactly are the forces (i.e. what are $\mathbf{f}_i$?) at various points on the cylinder (gravity, reaction force, etc.)?
Thank you for any help.