When some objects are plucked or hit, they vibrate at a particular set of frequencies. If you shake an object, or otherwise make it vibrate at one of those frequencies, it will start to vibrate more and more, often violently enough to break.
- Push a marshmallow onto the end of a piece of pasta.
- Break another piece of pasta so that it is approximately two-thirds the length of the first piece. Push another marshmallow onto the end.
For these explorations, you should have two pieces of spaghetti—one short, and one long—both with marshmallows on top.
To begin, hold the free end of the long piece of spaghetti in your hand. Using your other hand, gently pull the marshmallow a centimeter toward you and release. Observe the vibrations, and try to get a sense of how rapidly the marshmallow vibrates back and forth. The number of times something vibrates back and forth in a period of time is called its frequency, and when an object is allowed to vibrate on its own accord, that’s called its natural frequency.
Dampen the vibrations of the spaghetti with your free hand. Now start shaking the long spaghetti again by moving your hand only a small distance back and forth, starting with only a few shakes per second (a low frequency). You are making the spaghetti to vibrate at the same frequency as your hand. Making an object vibrate is called a forced vibration.
Slowly increase the number of shakes per second, and force the spaghetti to vibrate at a higher frequency. The marshmallow will probably twitch but its motion will remain relatively small at first. As your hand’s frequency goes up, though, at some point the spaghetti will suddenly start to vibrate a much greater distance back and forth, even though your hand is only moving a small distance back and forth. When an object is forced to vibrate at the same frequency as its natural frequency, it’s called resonance.
When an object is in resonance, a series of small vibrations can add up to create a larger vibration. Look at how much your hand is moving compared to how much the marshmallow is moving. Your hand is barely moving, but the marshmallow is moving a lot. Can you vibrate the marshmallow enough so that the spaghetti breaks with only a small motion of your hand?
Repeat the above steps with the shorter spaghetti. Does the shorter spaghetti-and-marshmallow arrangement have the same natural frequency as the longer one? Do you have to vibrate your hand at the same frequency to make resonance occur or do you need to shake your hand with a higher frequency?
Try the two spaghetti sticks at the same time. Hold both in one hand so that they make a “V”. Start shaking your hand. Can you get both marshmallows to move a large distance at the same time? That is, can you get both pieces of spaghetti to be in resonance at the same time?
You can force an object to vibrate in a variety of ways. You can touch it with the handle of a tuning fork. You can sing at it. You can rub a bow over it like in a violin. You can even buzz your lips at it like in a trumpet. Or, perhaps most obviously, you can grab it with your hand and shake it. But that’s not the only way an object can vibrate. Many objects have a natural frequency, which means that, when pulled or displaced from their initial position, they will spring back and vibrate for a time by themselves. Guitar strings, tuning forks, wine glasses, buildings, tubes of air, and many other things will vibrate at particular frequencies if part of the object is pulled back and then released.
Something amazing happens if you combine both of these situations together. If you force an object to vibrate at its natural frequency, it will start to vibrate at a very large amplitude. To visualize why, imagine the spaghetti and the marshmallow swinging back and forth. If you were to push the spaghetti with your finger at just the right moment—when it’s just starting to swing away again—the marshmallow will go just a little bit farther than it did the last time, getting a bigger amplitude. If you push on the spaghetti at the wrong time—such as when the marshmallow is swinging back toward you—you’ll slow the marshmallow down instead of speeding it up, and its amplitude will be less than it was before. Therefore, to make the marshmallow swing back and forth a greater and greater distance, you have to push on the spaghetti in sync with how it’s already swinging. In other words, to make the marshmallow have a greater amplitude, you must force it to vibrate at its natural frequency. At this frequency, the smallest push can add up to a swing large enough to break the spaghetti.
Each piece of marshmallow-topped spaghetti has a natural frequency that depends on the stiffness of the spaghetti, how much mass the spaghetti and marshmallow have, and how far away each mass is. The longer or denser the pasta or marshmallow, the lower the natural frequency. The stiffer the pasta, the higher the natural frequency. Depending on these factors, each type of pasta will have its own natural frequency.
The short piece of pasta will have a different natural frequency than the long piece of pasta. When you put them in the same hand, you force both to vibrate with the same frequency. If the natural frequency of one piece of pasta matches that frequency, then that piece will resonate, but not the other. Since they have different natural frequencies, they need different frequencies of forced vibrations to make them go into resonance.
The mechanism in this Snack is essentially the same as pushing someone on swing. If you pull back on the swing and then let go, it will take a certain number of times per second for the person to swing back and forth (natural frequency). If you want them to swing higher (get a larger amplitude), you have to push at the moment they’re about to swing away from you. If you get the timing all in sync, they’ll start to swing higher and higher, which is resonance. Notice that your hands are only pushing a few inches, but the person is swinging many feet above the ground.
This Science Snack is part of a collection that showcases female mathematicians and math educators whose work aids or expands our understanding of the phenomena explored in each Snack.
Bridges need to stand up to wind, waves, and their own weight. Hilda Geiringer is responsible for the safety of thousands of bridges around the world. A Jewish mathematician and the first woman lecturer at the University of Berlin, Geiringer left Nazi Germany for New York in 1939. In America, she co-developed slip-line theory, which describes the point at which metal bends or breaks and also helps us plan ahead for safety during earthquakes. Explore how spaghetti bends or breaks in a simulated earthquake with our Spaghetti Resonance Science Snack.