There is a nice puzzle about reel of thread. I wonder why the solution works.
You take a reel with threads on it. Inner radius r, outer radius R. You put it on a table. If you pull the tread under big enough angle Alpha the reel will move in opposite direction. The question is what is the minimal Alpha angle required. No slippage and rolling friction are present.
The solution takes tangency point of the reel and table and says that it won't move. So friction force $F_f = F \cdot \cos(\alpha)$. Then it considers moment of forces about the center of the reel. If $F \cdot R > F_f \cdot r$ the reel will move in the opposite direction.
The solution is quite simple and clear. But there is one thing I can't understand. If $F_f = F\cos(\alpha)$ and both gravity and normal reaction forces are normal to the table then why does the centre mass of the reel move? The sum of projection of all forces is equal to zero, so it should not move. Where am I wrong here?