Built on Facts

A while back the county government in my home town was considering a request from a phone company to construct a new cell tower to bring cellular service to a rural area. The government balked, largely for aesthetic reasons, though they couched their opinion in safety terms. They said they were concerned about the tower falling in inclement weather. Judging by the location, the maximum possible damage might be to one of the poor cows in the field surrounding the site. But that’s politics for you. Modern tower engineering is very well developed and your average cell tower will not be brought down by much short of a direct tornado hit. But what happens when a tower does fall?

Make a nice tall tower out of something. Lego bricks, building blocks, Oreos, anything. Then let it topple over. If you pay particular attention, you may notice that the tower has probably failed to fall in one piece. Long before it hit the ground, it likely snapped. The taller the tower, the more likely this is to happen.

Why? Presumably some kind of bending force is involved, since that’s usually what snaps things in half. Let’s see if we can track down its source. Maybe we’ll get somewhere by calculating how fast the tower should fall under the influence of gravity.

There’s a couple ways to attack this problem, but as is often the case it’s easiest to look at things from an energy perspective. We’d like to know the gravitational potential energy of the tower as it’s standing before it falls. We might guess it’s an average between 0 and mgL where L is the full height, and that guess would be right. To show this, we can take each bit (mass per length λ) of the tower with potential energy gλx dx and integrate between the bottom and top of the tower, sure enough:

and when it hits the ground all that will have been converted into kinetic energy of rotation about the tower base. In that case, K = (1/2)Iω², where I is the moment of inertia and ω is the angular velocity - revolutions per second times 2π. Don’t worry about the 2π if you don’t know why it’s there. Just be assured that in a lot of problems it makes the math much easier to include it. The moment of inertia is basically just a measure of how much inertia a rotating object has. We can look it up in a table or calculate it in the following way (again, don’t worry about the details if you’re not familiar with the procedure)

Now use that in the rotational kinetic energy equation and set it equal to the gravitational potential energy.

And solve for ω, the angular velocity.

Ah ha! Disaster! If the tower wants to naturally stay in one piece, the whole thing needs to fall at the same angular speed. But it doesn’t! The taller the tower is (and thus the bigger L), the smaller ω is. This means taller towers will take longer to fall down - just what we expect. But that means the upper part of the tower has to hold back the lower parts of the tower from reaching the ground as quickly as they would naturally. This requires a force, and if the tower is long enough that force will break the tower.

With a little more effort a clever person can calculate where the tower will break. Here’s a site where some physics students from LMU did that calculation and experiment, which you may well find pretty interesting. Think about it the next time you see a smokestack!