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 $\map y t$ 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:

$\dfrac {\d^2 y} {\d t^2} = g \paren {l - x} \dfrac {\d^2 y} {\d x^2} - g \dfrac {\d y} {\d x}$


Proof


Source of Name

This entry was named for Daniel Bernoulli.


Historical Note

This problem was first discussed by Daniel Bernoulli around the year $1732$.

In $1781$ Leonhard Paul Euler took up the problem, and found a solution for the special case: $u \dfrac {\d^2 y} {\d u^2} + \dfrac {\d y} {\d u} + y = 0$


Sources