There is a door with length $r$ and mass $m$ hinged on a wall with angle $\theta$, and an equally distributed gust of wind with constant velocity $v$ pushing on that door in the direction normal to the wall, as shown in the drawing:

[![enter image description here](https://i.stack.imgur.com/MlhyW.png)](https://i.stack.imgur.com/MlhyW.png)

I wish to find a function of $\theta$ by time. This is what I made of it so far (as someone with no experience in physics):

According to [this](https://www.youtube.com/watch?v=kNL5w73owzU&t=90s) video, for a fluid acting normally on a wall, the force is:
$$F = v\_a^2\cdot \rho\cdot A$$
Where $\rho$ is the fluid density and $A$ is the area on which the fluid hits the wall. I have silently replaced $A$ with $L$ (representing length) in my calculations since the problem here is 2 dimensional instead of 3. However, I am not sure I can do this.
So since the wind is acting down, the force vector is: $ \newcommand\mycolv[1]{\begin{bmatrix}#1\end{bmatrix}}
$
$$\overrightarrow{F}=\mycolv{0\\-L\rho \cdot v\_a^2}$$
However the wind is not normal to the door. So taking the vector normal to the door, that is
$$\mycolv{\cos(\theta-\frac{\pi}{2})\\ \sin(\theta-\frac{\pi}{2})}$$
and taking the dot product with the force vector to get the part that is normal to the door, we get:

$$F\_n=-L\rho \cdot v\_a^2\cdot \sin(\theta-\frac{\pi}{2})$$
L is the projection of the door onto the wall, since the wind hits normal to the wall. Meaning:
$$F\_n=-r \cos(\theta)\rho \cdot v\_a^2\cdot \sin(\theta-\frac{\pi}{2})$$
Simplifying, we finally get:
$$F\_n=r \rho \cdot v\_a^2\cdot \cos^2(\theta)$$
So:
$$F=ma \Longrightarrow r \rho \cdot v\_a^2\cdot \cos^2(\theta)=m\ddot{\theta}$$
And we reach the ODE:
$$\ddot{\theta} = \frac{r \rho v\_a^2}{m}\cdot \cos^2(\theta)$$
Which, while to my knowledge cannot be solved using standard mathematical functions, provides a satisfying enough answer for me.

When I spoke to my physics teacher about this however, he approached it differently. First he said he didn't think my method would work, because "the moment changes with time". I assume he's referring to moment of inertia. He then approximated the motion of the rotating door with a point particle which was traveling in a straight line. He said this was a standard method of approximation, some term with the word circular. And while the answer he gave is (according to him) a good approximation, I would like to know if my "answer" is the completely correct one, and if not, what is.