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18.2: The Intrinsic Equation to the Catenary

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    7045
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    We consider the equilibrium of the portion AP of the chain, A being the lowest point of the chain (Figure XVIII.1).

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    It is in equilibrium under the action of three forces: The horizontal tension \( T_{0}\) at A; the tension \( T\) at P, which makes an angle \( \psi\) with the horizontal; and the weight of the portion AP. If the mass per unit length of the chain is \( \mu\) and the length of the portion AP is \( s\), the weight is \( \mu s\text{g}\). It may be noted than these three forces act through a single point.

    Clearly,

    \[ T_{0}=T\cos\psi \label{18.2.1} \]

    and

    \[ \mu s\text{g}=\ T\sin\psi, \label{18.2.2} \]

    from which

    \[ (\mu s\text{g})^{2} + T_{0}^{2}=\ T^{2} \label{18.2.3} \]

    and

    \[ \tan\psi=\frac{\mu\text{g}s}{T_{0}} \label{18.2.4} \]

    Introduce a constant \( a\) having the dimensions of length defined by

    \[ a=\frac{T_{0}}{\mu\text{g}}. \label{18.2.5} \]

    Then Equations \( \ref{18.2.3}\) and \( \ref{18.2.4}\) become

    \[ T\ =\ \mu\text{g}\sqrt{s^{2}\ +\ a^{2}} \label{18.2.6} \]

    and

    \[ s\ =\ a\tan\psi. \label{18.2.7} \]

    Equation \( \ref{18.2.7}\) is the intrinsic equation of the catenary.


    This page titled 18.2: The Intrinsic Equation to the Catenary is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.