(*High school/college level*)

The formulas *y=k x^2* and *y=k cosh(x/k)* for the shape of the
cables come from considering all of the forces acting on the cable at any
given point. We won't derive the formulas here, because they involve
solving **differential equations**. But it's fun to
figure out what the forces are anyway.

Let *P* be some point on a freely hanging cable. You can imagine
that if we knew what the angle *a* between the cable and the
horizontal was at every such point *P*, then we could draw the curve
exactly.

So that's what we're going to do: describe what the angle *a* is at
every point.

Let's focus on *P*. The forces in the rope at *P* are
tangent to the rope: there is a force *T* pulling up and to the
right, which is balanced by a force *U* pulling down and to the
left.

We will concentrate on the force *U*. We can break *U*
into its horizontal (left) and vertical (down) components, as pictured
above. The force pulling down is the force due to gravity -- in other
words, the force due to the weight of the part of the cable pulling
down on *P*. But which part? Certainly the portion of the rope
above and to the right of *P* isn't pulling at *P*. What
about the portion to the left?

If you think about it, you can convince yourself that it's only
the portion of the cable from *P* to the lowest point that is
pulling down at *P*. If you have a piece of heavy rope, you can
experiment by hanging it between your hands. As you raise your left
hand, you can feel the weight supported by your right hand decreasing.
At the lowest point, the rope is horizontal, which means that all
forces are pulling to the left and right, and there is no downward
component.

So, if *w* equals the the weight of the rope per unit length, and
we let *s* be the length of the portion of the rope between the lowest
point and *P*, then *P* feels a force of *ws* pointing
straight down.

Now what about the force pulling to the left? The only horizontal
force anywhere on the rope is the tension due to stretching the rope
between the pylons. Since this is the only horizontal force, and
since the rope is in static
equilibrium, every point on the rope feels the same horizontal
tension *H*. We could measure *H* at the lowest point,
where there are no other forces acting.

So the force vector *U* has horizontal component *H* and
vertical component *ws*. These must match the horizontal and
vertical components of *T*.

In terms of the angle *a* we have

Notice the tension *T* in the suspension cables increases as s
increases. The tension is maximum at the top of the pylons, where the
cable is supported.

The two equations imply that

We've done it. This equation describes the angle of the curve at
each point, as a function of the distance s from the lowest point.
Both *w* and *H* are constants. The constant *w* is
just a property of the cable, and the tension *H* at the lowest
point is something we can control -- as we take up the slack in the
cable, *H* increases.

Exercise 1: Use what you know about the tangent function to
describe how *a* changes when we change *w* and *H*.
For example, if the weight per unit length *w* is fixed, and the
tension *H* is increased, how does *a* change at a fixed
distance s along the cable? Describe how this changes the shape of
the catenary.

The quantity tan*(a)* is telling us the slope of the tangent
line to the cable at each point. This is a concept central to
**calculus**, and knowing some calculus would enable us
to use the above equation for tan*(a)* to see that the equation of
the curve described by the cable is

What changes when we hang a heavy road from our suspension cable?
Not much, really! The only thing that changes in the above discussion
is the weight pulling down at the point *P*. We will assume that
the weight *W* per unit length of road is **much**
heavier than the weight *w* per unit length of cable. In that
case, the weight supported at *P* is essentially *Wx*, where
*x* is the **horizontal distance** from the lowest
point, instead of the distance *s* along the cable.

Exercise 2: Use this to show that the equation for *a* becomes

Thus the equation describing *a* is a function of *x*
instead of *s*. It leads to the equation

for the cable supporting a heavy road.

Back: Equations for the Shape of the Bridge.

Return to the Golden Gate Bridge.

Last modified: Mon Sep 16 14:03:27 1996