MHB Vibrations of a Hanging Chain: Modeling Tension with PDEs

AI Thread Summary
The discussion centers on modeling the vibrations of a hanging chain using partial differential equations (PDEs). The key point is to derive the tension at a point along the chain in equilibrium, expressed as τ(x) = ρ · g · x, where ρ is the mass density and g is gravitational acceleration. The equilibrium condition considers the balance between the downward gravitational force and the upward tension. Participants are encouraged to build on this foundation to further analyze the system. Understanding these forces is essential for solving the problem effectively.
cbarker1
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Dear Everyone,

I am having trouble with how to start with one part of the question:
"In this exercise, we derived the PDE that models the vibrations of a hanging chain of length $L$. For convenience, the x-axis placed vertically with the positive direction pointing upward, and the fixed end of the chain is fastened at $x=L$. Let $u(x,y)$ denote the deflection of the chain, we assume is taking place in $(x,u)$-plane, as in the figure, and let $\rho$ denote its mass density (mass per unit length).
(Here is where I have trouble starting)
Part a:
Show that, in the equilibrium position, the tension at a point $x$ is $\tau(x)=\rho \cdot g \cdot x$, where g is the gravitational acceleration.

Thanks,
Cbarker1
 
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Hi Cbarker1,

In the equilibrium position the chain is hanging straight down. At a point $x$ along the chain there are two forces we must consider: (i) the weight of the chain pulling the point down via gravity and (ii) the tension at $x$ pulling up. The equilibrium equation is then $$\tau(x)-mg=0,$$ where $m$ is the mass of the portion of the chain pulling down at the point $x$. Can you continue from here?
 
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