Monday, July 21, 2008

Answer: Suspension Bridge
New Problem: Inverted Pendulum

Suspension Bridge
The cable takes the shape of a parabola. Consider the section of cable supporting a differential element of the road, length dx.

The physics that goes into this solution is:
  • the total force on this bit of cable is zero
  • the road pulls down on the cable due to its weight from gravity
  • the tension of the cable pulls from either side; this tension must be along the direction of the cable
We'll break the tension on each side into its x and y components. The x-components must cancel each other because they're the only forces in the x-direction. The equation stating that the total y-forces all cancel could be written

That last term is the linear mass density of the bridge times gravitational acceleration times the length of the segment. All together it's the weight of the road supported by that segment of cable. Finally, we bring in the fact that the tension must be along the direction of the cable

Now just mop it up

The equation gives a parabola. The constants "c1" and "c2" simply allow you to change the focus of the parabola. This is equivalent to saying that you are free to move the origin of the coordinate system around as you wish without changing the shape of the bridge. The third constant, Tx, does change the shape of the bridge. If you make Tx large, you can have a shallow parabola with low towers supporting the bridge, but you'll need a strong cable capable of supporting a lot of tension. On the other hand, you could get away with a weaker cable by making Tx small, but then the parabola would open up steeply, and you'd need a very long cable and tall towers to make it work.

Inverted Pendulum
The problem is basically, "how does this work"?

We went to the Exploratorium this weekend, and one exhibit was a pendulum attached to a slider bar that ran on a horizontal track. The idea was to slide the bar back and forth in such a way as to swing the pendulum to the vertical, then try to keep it there. Many people could get the pendulum swinging half way up, but then were stuck swinging the slider back and forth violently without getting the pendulum to rise any higher. However, it's possible to swing the pendulum around without using much force, without swinging the slider long distances, and done completely with your eyes closed. How? What algorithm would you use?

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