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A bad drawing just so you can see what I mean: enter image description here

On every derivation I see, at some point it is stated that: $$ \frac{T_x}{T}=\frac{-x}{L}$$ It is always explained by saying "by the geometry of the problem" or "by triangle similarity" but I fail to see how that follows so easily. This relation implies that as the x component of the mass gets larger, more tension is in that axis. That of course makes sense, but how is it possible to explicitly see this geometrically?

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It actually is straightforward geometry if you draw the figure properly. Look at $\theta$, the angle made by the pendulum with the $zy$ plane. It’s cosine is the required ratio.

enter image description here

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  • $\begingroup$ And where is the minus x come from ? Because Tx $\endgroup$
    – Eli
    Aug 20, 2020 at 21:11
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    $\begingroup$ $\hat{x}$ and $\vec T_x$ point along opposite directions. $\endgroup$ Aug 20, 2020 at 21:12

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