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# Explores the Conditions for Stability and the Idea of Center of Mass

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Source:
Author: Jerry Silver

### The Idea

How far can an object tilt before it falls? Like the Leaning Tower of Pisa, the stability of a rectangular or cylindrical object depends on its shape. This experiment establishes a simple condition for stability of an object and explores the idea of center of mass.

### What You Need

• cereal box
• pizza box
• 2 pencils
• tape
• string
• 2 nuts, large washers, or other matched attachable weights
• wooden board to use as an incline (roughly 3 ft × 4 inches × ½ inch, or 1m × 0.1m × 0.01m)

### Method

1. Find the center of each of the rectangular faces of the box.
2. Start with the largest face first. Push one pencil through both sides of the box. The pencil should be roughly perpendicular to the surface it is pushed through.
3. Tie the string—one end to the pencil and the other end to the hangable weight.
4. Attach the other weight to the other side of the pencil as a counterbalance. You can use string if that makes this easier.
5. Tape the other pencil across the incline, somewhere roughly near the midpoint.
6. Place the box on the incline, so the downhill side of the box is in contact with the pencil taped to the incline. This pencil serves as a pivot point to force the box to rotate, rather than slide down, the incline.
7. Make your predictions. How far can you lift the incline before the box topples?
8. Try this with the various faces of each of the boxes. Can you develop a general condition for stability?
9. You can do this qualitatively as discussed previously or take it a step further and relate the geometry of the box to the angle it can tilt at and still be stable. Can you predict the maximum angle of stability for given box dimensions?

### Expected Results

The box will be stable if the center of mass (marked by the pencil) is over the base of the box in contact with the incline. Once the angle increases to the point where it is outside the base, the object will rotate.

Objects are more stable when the center of mass is closest to the incline.

Because a pizza box has a square-top face, it will be stable up to a 45-degree angle when propped up with one of the long edges placed along the incline, as shown in Figure 47-1.

If A is the length of the side of the box in contact with the incline and if B is the height of the box (for that particular arrangement) above the incline, the maximum stable angle is given by: tangent (angle) = A/B. (The angle can be found by taking the inverse tangent or arctan, which can be found on most scientific calculators.)

For instance, a 17-ounce box of Honey Nut Cherrios has dimensions 12 inches × 7 ¾ inches × 2 ¾ inches. The six possible placements for this box are summarized in Table 47-1.

A few of these are illustrated in the following Figures 47-2, 47-3, and 47-4.

### Why It Works

Massive objects tend to act as if all their mass was concentrated in a single point called the center of mass. Gravity pulling on that point causes the box to rotate about the pivot point established by the pencil. If the center of mass is above the base of support, the object tends to rotate in such a way as to remain stable on the incline. However, as the center of mass moves out from above the base, a torque is applied, which tends to rotate the object, so it rolls down the incline.

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