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Dialogue of Cultures

# Pedagogical Studies:

"Justice for the Catenary"
Diagrams

### by Bruce Director

#### FIGURE 1

The shape the chain assumes when hanging under its own weight, is that shape which equalizes the tension at every point. That is, any one link will have exactly the same amount of pull from either side along the chain. Links near the top of the chain support much more weight than links near the bottom, yet they don't have to made of stronger material than those at the bottom. A link at point A has much more chain pulling it to the right than to the left, yet it doesn't move. The position of each link in the chain, is thus, a function of these physical principles.

#### FIGURE 2

2(a) Varying the lengths of the chain generates a family of catenaries of varying curvatures.
2(c) Release catenary AB to points C,D. Every arc of a catenary, is itself a catenary!
2(b) Varying the endpoint of a fixed length of chain generates a second family of catenaries.

FIGURE 3
The force exerted at positions A and C of the hanging chain, is the same as if an object whose weight was equal to the weight of the chain hanging between those points, was suspended by threads that were tangent to the catenary at A and C.

#### FIGURE 4

In 4(a) the relationship of the force exerted at A to the force exerted at C is proportional to the sine of angle ABG to the sine of angle CBG. The following demonstration illustrates this principle intuitively. Have two people hold a weight suspended by two strings as depicted in 4(b). It will take a certain amount of effort on the part of both people to lift the weight vertically. Now, have the two people gradually step back from each other, while continuing to hold the weight up at the same height as depicted in 4(c). The effort required to hold up the weight increases as the distance between the people increases. The rate at which this effort increases is proportional to the sines of the angles the strings make with a vertical line drawn through the weight. 4(d). Try experimenting with different angles and notice how the effort required to hold the weight up changes. For example, if person A holds his end of the rope higher, thereby changing the angles, the effort both he and the person at B have to exert to hold up the weight also changes. 4(e)

#### FIGURE 5

The force exerted on the lowest point doesn't change regardless if the chain is increased or decreased. If the chain is increased from F to A in 5(a) the force at B is unchanged.

#### FIGURE 6

6(a) The catenary is the path that maintains an equal force on the lowest point. In the experiment of two people holding a weight by two strings, the catenary is the path person A would have to follow so as not to change the force required to lift the weight, while person B remained stationary. As person A moves away from B, he has to also hold the rope higher to keep the force constant. In other words, the person at A has to move non-uniformly vertically with each horizontal movement, in order to keep the force exerted in lifting the weight unchanged.
6(a) The catenary is the path that maintains an equal force on the lowest point. In the experiment of two people holding a weight by two strings, the catenary is the path person A would have to follow so as not to change the force required to lift the weight, while person B remained stationary. As person A moves away from B, he has to also hold the rope higher to keep the force constant. In other words, the person at A has to move non-uniformly vertically with each horizontal movement, in order to keep the force exerted in lifting the weight unchanged.
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