The Hanging Chain

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St. Michael the Archangel defend us in battle. Be our protection against the wickedness and snares of the Devil. May God rebuke him, we humbly pray, and do thou most Heavenly host, by the power of God cast into hell Satan and all the evil spirits who roam around the world seeking the ruin of souls. Amen.

Solving the physics problem pertaining to the hanging chain. (Krantz)

* .... Differential Equations for solving physics problems.

* .... have method that we know to solve these equations.

So, what is the classic Hanging Chain formulation. What is all that mess below? Well imagine a metallic chain hanging in the middle under the influence of gravity. In architectural terms it's a very important structure and can be mathematically described. It is in fact related to the arch, but upside down. We analyze points A (the midpoint of gravitational attraction) and B (an arbitrarily set point) on the chain. We also apply at point A T1, tangential point of tension at A. And at B we apply T2 the tangential component of tension at that point at an angle Theta with respect to the x-axis. In the y direction there is also a component of T2 as well as w, the weight of the chain at A.

What we next must do is break the tension into components using Mechanics/Statics. Thus, in the x-direction we have T1=T2cos(angle), while in the y-direction ws=T2sin(angle) where ws is the weight of an incremental arclength s as can be seen in the diagram. Dividing the first equation by the second equation to get ws/T1 = tan(angle).

Now taking y' = tan(angle) = ws/T1 (due to the fact that y(x) is a graph of the chain) we can continue with the analysis. Using the arclength delta s as the hypotenuse we continue with the application of the Pythagorean theorem. This leads to differential equation that must be integrated after plugging in values from the previous section.

After getting the integral value from the table and using the given initial condition y'(0) = 0 we find that C = 0. Next we take hyperbolic sin (sinh) of both sides and solve for dy and finally integrate to get y'(x)!

Using initial conditions for y(0) = 0 we then integrate and find the value of D. We conclude the process finally finishing with the differential equation below.

Let us move on now !!! War Strategy!!!

Now moving onto Pursuit curves...Pursuit Curves

Go back to Gauss' Law Problem from here!

1st Gauss' Law Example problem ... Gauss Law Example 1.

Navigate below to go back to the start!

You can go to DE Ex2 here...D.E. Ex2.

You can go to DE Ex1 here ...D.E. Ex1.