# Physics and the Tower of Pisa

## Objective

To learn about stable equilibrium and center of mass through the Leaning Tower of Pisa.

Procedure: Easy

Concept: Medium

## Concept

If you have a minute, try this experiment: Get an unopened can of pop and try to balance it on its edge (see picture below).

Having difficulty? Well open the pop, drink about half of it and then try balancing (without spilling). Now you should be able to do it. Still thirsty? Finish drinking the pop and see if you can balance it now; I doubt you'll be able to. This simple experiment motivates a study of center of mass and stable equilibria.

Imagine now that instead of a pop can you have a tower, and that you are trying to balance that tower on its edge in soil. It's obviously a much bigger problem for a tower to fall over than half a can of pop.

Such a problem is much more complicated and has burdened Italians for many years now. To prepare for this project you should research the Leaning Tower of Pisa and the attempts made to stabilize it. Also study the concepts of center of mass and perhaps stable equilibrium to have a basis for why an object stands up or falls.

## Materials

• Several pieces of cardboard
• Duct tape (or glue or staples)
• Ruler for metric measurements
• Protractor for angle measurements

## Procedure

1. Using the cardboard and duct tape (or other attachment) you have, build a cylinder. You can make the dimensions anything you want (as long as the cylinder is taller than it is fat.
2. As a reference, the real Tower of Pisa is 55 meters high and 7 meters in diameter. If you used a 12 cm base, your cylinder would have to be about 95 cm tall to match the tower.
3. Measure the height of the cylinder and the diameter of the base for whatever tower you end up making. This doesn't have to be a perfect cylinder, but the bottom needs to be flat so that the cylinder stands straight up. It might look like this, although this is a bit small:
4. An outline was used to cut out the pieces of this cylinder.
5. Take some of the leftover cardboard and cut a couple pieces out so that they can be stacked.
6. Put one piece of cardboard under your tower and see if it stays standing. Measure the angle at which it the tower is tilting. See the picture below for a demonstration of which angle to measure.
7. Put additional cardboard pieces under the tower, continually increasing the angle until your tower falls over. Try to determine the exact angle at which your tower falls over. Record that angle along with the dimensions of your tower in a table.
 Height (cm) Base (cm) Max Angle (°) 55 7 ? ... ... ...

## Analysis

The concept in Physics called center of mass tells us that the mass of a cylinder (tower) is at its center. Hopefully your research revealed this to you. A tower is at equilibrium when its center of mass is above its base; see the moving picture above to illustrate this. Logically then, the tower will fall over when its center of mass is not above the base. Using trigonometry and geometry, determine the angle at which the tower should fall. How do your experimental results compare to these analytical results?

## Extensions

• Try this experiment with a rectangular prism, or a triangular prism (or any cylinder who's base is not a circle). How do your results compare between a triangular prism and your original tower? Consider that this difference may be because the center of mass is different for different shaped bases.
• What if you tried to use a cylinder which is fatter than it is tall? Try the experiment again with a cylinder that is 5 cm tall and 25 cm wide. How big does the angle have to be? Does this still conform to the idea of center of mass?
• If it is possible, try to consider using a filled cylinder instead of a hollow one as you have been using. Use a filled cylinder and a hollow cylinder of the same size and compare the angles of collapse. Are they the same? What does this tell you about the location of the center of mass for a filled cylinder?