I was putting my belt to hang on the clothes rod which promptly fell with a loud clang. Obviously, I had left too much of the belt hanging on the buckle side.
I smiled as I remembered the pulley problems I used to solve. It reminded me of the typical free body diagrams we used to try. So if I had to find out how much of the belt I need to leave hanging on both sides of the rod, I could reduce it to an ideal problem by replacing it exactly as the above string pulley system with the mass of each side replaced as a point mass at the centre of mass.
Hence, its clear why my belt fell. At a point, the mass on the side with the buckle would be too heavy (have a large mg component) to be compensated by the other.
I assumed that both sides of my belt were at a state of equilibrium usually and that's why it stayed stationary. After a while I started playing with it and something kept nagging me. The buckle side of my belt fell down much later than I would have anticipated considering how light the rest of the belt was.
Then, it hit me. This couldn't be solved as an ideal pulley system cause I had forgotten one very important assumption.
The pulley is rigid and frictionless.
Friction. That was what was saving me from having bother about belt lengths usually. As long as the mg components remained less than the co-efficient of static friction between my rod and the belt, it was able to satisfy a large amount of stress. It was only when it crossed the limiting friction, it fell.
Ah, I love physics. I maybe wrong, these are just random thoughts based on my understanding and you have been warned, this is no guide to physics, just a flavour of my thoughts.