Bernoulli's Hanging Chain Problem
Theorem
Consider a uniform chain $C$ whose physical properties are as follows:
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
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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
- 1922: Andrew Gray and G.B. Mathews: A Treatise on Bessel Functions (2nd ed.) ... (next): Chapter $\text{I}$: Introductory: $\S 1$. Bernoulli's Problem