Modeling a hanging chain as a PDE

[nettle404]

Homework Statement

A flexible chain of length $\ell$ hangs from one end at $x=0$ but oscillates horizontally. Let the $x$ axis point downwards and the $u$ axis point to the right. Assume that the force of gravity at each point of the chain equals to the weight of the part of the chain below the point and is directed tangentially along the chain. Assume that oscillations are small. Find the PDE satisfied by the chain.

Homework Equations

I will liberally use Newton's law $\textbf F=m\textbf a$. I will use $g$ as the gravitational acceleration.

The Attempt at a Solution

The longitudinal component of tension must counteract gravity, hence

$$\frac{T}{\sqrt{1+u_x^2}}=-\int_x^\ell\rho g\,dx'=\rho g(x-\ell)$$

Observing that oscillations are small, we have $\sqrt{1+u_x^2}\to1$, and so $T=\rho g(x-\ell)$. The transversal component of tension at any given $x$ is

$$\frac{Tu_x}{\sqrt{1+u_x^2}}=\int\rho u_{tt}\,dx$$
$$\frac{\partial}{\partial x}\frac{\rho g(x-\ell)u_x}{\sqrt{1+u_x^2}}=\frac{\partial}{\partial x}\int\rho u_{tt}\,dx$$

Once again using the small oscillations to remove the denominator:

$$\rho gu_x+\rho g(x-\ell)u_{xx}=\rho u_{tt}$$

Which is my PDE. Is this correct?

 

[mighty2000]

Your approach looks correct, but there are a few things to note:

1. It would be helpful to define your variables more clearly. For instance, what does u represent? Is it the displacement of the chain from its equilibrium position? And what is x? Is it the horizontal position along the chain?

2. Your first equation, T = ρg(x-ℓ), is correct only if the chain is in equilibrium. In this case, the weight of the chain must be balanced by the tension in the chain. However, in this problem, the chain is oscillating, so the tension will vary along the chain and will not be constant at every point.

3. Your second equation, Tux/√(1+ux^2) = ρu_tt, is also correct, but again, it only applies if the chain is in equilibrium. In this case, the horizontal component of the tension must balance the acceleration of the chain. However, since the chain is oscillating, the horizontal component of tension will not be constant at every point.

4. The PDE you have derived is the wave equation, which is valid for small oscillations of a string or a chain. However, the problem statement explicitly states that the oscillations are small, so this is not a problem. Just be aware that this PDE may not be valid for larger oscillations.