Bernoulli's Hanging Chain Problem

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Theorem

Consider a uniform chain $C$ whose physical properties are as follows:

$C$ is of length $l$
The mass per unit length of $C$ is $m$
$C$ is of zero stiffness.

Let $C$ be suspended in a vertical line from a fixed point and otherwise free to move.

Let $C$ be slightly disturbed in a vertical plane from its position of stable equilibrium.

Let $y \left({t}\right)$ be the horizontal displacement at time $t$ from its position of stable equilibrium of a particle of $C$ which is a vertical distance $x$ from its point of attachment.


The $2$nd order ODE describing the motion of $y$ is:

$\dfrac {\mathrm d^2 y} {\mathrm d t^2} = g \left({l - x}\right) \dfrac {\mathrm d^2 y} {\mathrm d x^2} - g\dfrac {\mathrm d y} {\mathrm d x}$


Proof


Source of Name

This entry was named for Daniel Bernoulli.

He solved this problem around the year 1732.


Sources