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In my textbook, with $S=|\mathbf{S}|$, from the picture below, they derive

$\int_{x}^{x+h}\rho_lu_{tt}''d\lambda=S(x+h)\sin(\alpha(x+h))- S(x)\sin(\alpha(x))+\int_{x}^{x+h}Fd\lambda$, where $F$ is [N/m], $\rho_l$ is [kg/m] and $u$ the position in y-axis of a point of the string.

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Why is it though, that they put a minus sign between the internal forces of the string in the expression above? I understand the concept (except this addressed confusion) and further steps of differentiating (dividing by $h$ and limit) in other derivation, although what if the tangential force $S(x)$ is not facing in negative direction, but in a positive one? - then the expression would be fault? My textbook says that it is general.

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    $\begingroup$ The two force must be in opposite direction in order that there be tension in the string. If both tangential forces were positive (or both negative) there would be no tension- the string would just move in the direction of the forces! $\endgroup$ – user247327 Mar 20 at 12:59
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    $\begingroup$ Look at it this way: at each point of the string, the upper part pulls upward and the lower part pulls downward. When you consider your short fragment, you think of it as the upper part when you look at one end, and as the lower part when you look at the other end. $\endgroup$ – Ivan Neretin Mar 20 at 14:41

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