Given a horizontal stick AB and a string, of course a stick that is hang on the string in its center of mass is in equilibrium. This is a fact that we take as rule because we can observe it, right? I mean, this is not a fact that we can deduce by maths. Now, if you have for example two external and vertical downward forces, say $\vec F_1$ and $\vec F_2$. The first acts on $A$ and the second on $B$. Can we take as rule of observation that in order to have equilibrium we have to put the string on the point given by $d_1:d_2=F_2:F_1$ ? Or can we deduce it by maths?
Answer
Given a horizontal stick AB and a string, of course a stick that is hang on the string in its center of mass is in equilibrium. This is a fact that we take as rule because we can observe it, right?
We never can understand what is the fact. Because we never can discover true laws of physics (or nature) unless we ask the creative of the nature and as far as I know, no one has seen him. Hence, I correct your sentence as below:
This is a fact that we take as rule because we hope it be.
No one cannot prove laws of physics like laws of Newton for instance. What we can do is acceptance or rejection of them. We have accepted that if the net force acting on a body with constant mass is $\vec F$ then its acceleration will be $\vec a=\frac {\vec F}m$. We cannot prove or even observe it but we hope that it is true and we build machines, airplanes, etc. by it without certainty of its correctness. This is why human is a wonderful creature! He lives with things that never knows their correctness.
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