We want to attach n various weights to the perimeter of a circular turbine, equidistant from each other.
I thought about using the cosine function of the position, the weight and the radius somehow (maybe just simple multiplication), which would give us some sort of directional force (not a physics major, sorry I'm completely butchering this) and then we would maybe add that up for all the weights to get the total force/center of gravity? I don't completely understand that theory in itself, so even if it's somehow correct, which I doubt, please tell me why. And other than this, I have no clue.
Thank you
Let's recall the definition of the gravity center $\vec{r}_G$ for a system of $N$ point masses : $$ \vec{r}_G = \frac{1}{M}\sum_{k=1}^Nm_k\vec{r}_k $$ where $m_k$ are the individual masses, $\vec{r}_k$ their positions and $M = \displaystyle\sum_{i=k}^Nm_i$ the total mass in the system. Given that the weights are positioned equidistantly on the perimeter of a circle of radius $R$, one has : $$ \vec{r}_k = \begin{pmatrix} R\cos(k\theta) \\ R\sin(k\theta) \end{pmatrix} \quad\mathrm{with}\quad \theta = \frac{2\pi}{N}, $$ hence $$ \vec{r}_G = \frac{R}{M}\sum_{k=1}^Nm_k\begin{pmatrix} \cos(k\theta) \\ \sin(k\theta) \end{pmatrix} $$ which cannot be simplified further, because the weights aren't identical.