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Whirligigs

In this quiz, we're going to dissect the behavior of a common toy known as the whirligig, a handheld rotor that can fly several meters into the air before it runs out of energy. You may have one around the house—they're commonly given away as consolation prizes at birthday parties and arcades.

In explaining the whirligig, we'll also come to understand the basic principle of helicopters, and the maple tree leaf that slows its fall to Earth by spinning, increasing the range of seed dispersal. To start, we'll throw some bouncy balls at boards.

Suppose we shoot bouncy balls at the flat board, as shown below. During the collision, the balls exert a force on the board. In which direction is this force?

Suppose we shoot bouncy balls at the tilted board—which makes an angle of \(\ang{45}\) with the vertical as shown below—in what direction do the balls travel after the collision?

In the previous problem, what direction is the impulse force on the board?

We've shown that shooting bouncy balls at a wooden board accelerates the board in the direction perpendicular to its surface. Thus, by setting the angle of the boards surface, we can control where it is headed. At this point, you might be wondering "that's nice, but what does it have to do with the whirligig?"

The connection is that molecules of air act roughly like the bouncy balls we considered. When the rotor moves through the air, it is constantly colliding with air molecules and therefore receiving tiny impulse forces perpendicular to the orientation of the blades. By moving the rotor quickly, and making the rotor long, this sustained impulse force is enough to lift a helicopter.

Now let's look at some whirligigs.

Suppose we drop the whirligig shown below, which has two flat wings. In which direction will it move?

Now we drop the whirligig shown below, which has both rotors pitched at the same angle. How will it move?

We've seen how the interaction of the whirligig with the air can result in different movements. In one case we slowed its descent as it fell straight down, and in another the whirligig fell to the side as both rotors were accelerated in that direction. Now we have everything we need to understand how the whirligig (and a helicopter) flies into the air.

But first, let's drop one.

Suppose we pitched the rotors in opposite directions (as shown below), what will happen when we drop it?

When we drop the conventional whirligig (with the rotors cross-pitched), it spins on the way down due to torque about the center. As the rotors are spinning, they travel more quickly relative to air than if they were simply falling. This leads to stronger impulse forces, and therefore more force in the upward direction.

This suggests that if we can somehow increase the speed of rotation of the whirligig, we can increase the vertical impulse until it overcomes the pull of gravity. In that case, the whirligig wouldn't simply fall more slowly, it would take off. In fact, this is the principle underlying a helicopter. Because the engine can drive the rotor at high speed, the rotors can receive strong impulse forces on a continual basis, and thus hover for as long as they have fuel to run the motor.

Even nature has taken advantage of this mechanism (in a slightly different iteration). The seeds of maple trees are formed in such a way that a lightweight, pitched rotor is attached to a heavy core. The rotation of the seed as it falls leads to a significantly longer hang time, allowing whatever breeze there may be carry the seed farther, thereby enhancing the dispersal of maple trees.

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