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?
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$\begingroup$ What a real stick does can only be found out by observation, what theoretical forces acting on a theoretical stick do can be deduced mathematically. $\endgroup$– CuriousOneCommented Jun 27, 2016 at 20:28
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$\begingroup$ yes, the equilibrium point can be deducted theoretically $\endgroup$– user65081Commented Jun 27, 2016 at 21:50
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$\begingroup$ What does your physics text book say about this? $\endgroup$– sammy gerbilCommented Jun 28, 2016 at 0:21
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$\begingroup$ Sorry but I know that if we write the equations for the equilibrium we find a point $P$ on the stick where I should put the string. But we find also $\vec T=\vec F_1+\vec F_2$ (do not consider the weight of the stick). We do not have control on $T$, and suppose I give you a very robust string: what guarantee to us that if I put the string in $P$ then the tension will be exactly $\vec F_1+\vec F_2$? This is something that our experience suggests right? I mean it's like the fact that if I put an object on a very strong table it will be in the equilibrium right? $\endgroup$– RichardCommented Jun 28, 2016 at 11:25
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$\begingroup$ I mean, how can we predict the behavior of the tension of the string? $\endgroup$– RichardCommented Jun 28, 2016 at 12:09
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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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$\begingroup$ ok but once we have accepted laws like the conditions of equilibrium, why should we accept that putting the string in $P$ implies $\vec T=\vec F_1+\vec F_2$? Again, we accept this because of many observations? I mean, in general we study an object that we can observe and we base the equations on it. Here, for example, I'd say : I see the situation of my question and everything in in equilibrium. So I write equations for the equilibrium. These give to me that the point $P$ must be $lF_2/(F_1+F_2)$. $\endgroup$– RichardCommented Jun 28, 2016 at 13:07
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$\begingroup$ But what about the converse? If I have a stick and a robust string and someone tells me: Put this objects in equilibrium, everybody would put the string in $P=lF_2/(F_1+F_2)$, but this does not imply that $\vec T=\vec F_1+\vec F_2$ mathematically. So why everybody would be sure about equilibrium? $\endgroup$– RichardCommented Jun 28, 2016 at 13:07
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$\begingroup$ I say, they aren't sure! They hope and have accepted that stick will be in equilibrium if they put the string in $\frac{lF_2}{F_1+F_2}$. In addition, equations of the equilibrium that we have accepted, are $\Sigma \vec F=\vec 0$ and $\Sigma \vec M=\vec 0$. We have accepted both of them. $\endgroup$– lucasCommented Jun 28, 2016 at 13:29
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$\begingroup$ ok but why they accepted that stick will be in equilibrium if they put the string in $lF_2/(F_1+F_2)$? Only from this we don't have a $\textbf{mathematical proof}$ that $\vec T=\vec F_1+\vec F_2$. We can deduce the tension only if we $\textbf{observe}$ the objects. So are they sure because of their experiences? $\endgroup$– RichardCommented Jun 28, 2016 at 13:35
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$\begingroup$ Note: none of their accepted laws imply, $\textbf{without observing}$, that $\vec T=\vec F_1+\vec F_2$. $\endgroup$– RichardCommented Jun 28, 2016 at 13:42