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I'm working on an engineering problem about buckling of hydraulic cilinders. The relevant standard provides the following equation: $$\sqrt{\frac{I_1}{I_2}}*tan(\omega_1*L_1)+tan(\omega_2*L_2) = 0$$ where $\omega_1 = \sqrt{\frac{N}{E*I_1}}$ and $\omega_2 = \sqrt{\frac{N}{E*I_2}}$

$I_1$, $I_2$, $E$, $L_1$ and $L_2$ are all constants, which leaves N as the variable. The equation can be rewritten as $$\sqrt{\frac{I_1}{I_2}}*\tan\left(\sqrt{\frac{N}{E*I_1}}*L_1\right)+\tan\left(\sqrt{\frac{N}{E*I_2}}*L_2\right) = 0$$

With N as the variable, the task is to find the smallest positive root for this equation. Now my question:

Do I need to consider both the positive and negative outcomes of $\omega_1$ and $\omega_2$? Since the square root of any real number can have a positive and a negative result. Or am I thinking way too hard on this?

I don't need a detailed explanation on how to find the root, since I'm using a graphical method to approximate the root.

Ataulfo
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Dolf
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  • I don't think this is a math question...the scientific context ought to tell you whether the negative square roots are relevant. Note that the notation specifies the positive square root, so writing it the way you have indicates that only the non-negative roots are relevant. But of course we don't know that whoever wrote this formula was careful with notation. – lulu Dec 17 '24 at 11:39
  • The general consensus in mathematics is that the square root function returns the principal square root, i.e. the positive value, and only the positive value. Perhaps your standard clarifies if Sqrt is to be understood as a function or as a pair of numbers. – Randy Marsh Dec 17 '24 at 11:40
  • Furthermore, when one wants to emphasize that both positive and negative values are to be considered, they would typically write $x = \pm\sqrt{y}$. But there is no rule for this. This is probably a question/request for clarification to whomever is in charge of the standard that you're using. – Randy Marsh Dec 17 '24 at 11:43
  • Thank you both. I'm gonna go out on a limb here and assume only the positive results are relevant. – Dolf Dec 17 '24 at 12:14
  • This question asks about solving $\tan(nx)=k\tan(x)$ – Тyma Gaidash Dec 17 '24 at 14:48

1 Answers1

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$\sqrt x$ always means the non-negative square root of $x$. This is a well-established mathematical convention.

TonyK
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  • @Piquito: what are you talking about?? – TonyK Dec 17 '24 at 23:19
  • K: Simply that your answer could be just a comment. And because (I could be wrong) I believe that there are people in MSE who do not appreciate me and of which I believe I have very concrete proof. Anyway, I have nothing against you and I rather respect your person. Best regards. – Ataulfo Dec 18 '24 at 12:13