Key concepts
Physics
Resonance
Resonant frequency
Vibration

Introduction
Have you ever been on a swing set and suddenly noticed that the person on the swing next to you seems to be swinging almost exactly in time with you? You go up and down at either the same time or exactly opposite each other. This might seem random—but it's actually physics! Like many things in nature, swing sets have a resonant frequency, which means they have a “favorite” frequency (or speed) of movement. The swing set will naturally want to swing at its favorite speed. You might have experienced this if someone has ever tried to push you too fast on the swing; the preferred speed can actually make you go slower.

In this activity we'll use paper rings and (lots of shaking) to examine resonant frequencies for ourselves!

Background
If you've ever played a guitar, violin or other string instrument, you've seen resonant frequency in action. A single guitar string, when plucked, will vibrate at its resonant, or favorite frequency. The vibration of the string creates a sound wave, which we hear as a note. It's always the same note for the same string because that string (when tuned correctly) always vibrates at its resonant frequency.

Resonant frequency is determined by a several factors, including the mass of the object and the stiffness of the object. Again, if you've ever played a guitar, you may have noticed that the strings aren't all the same. The low E string is much thicker than the high E string; because it is thicker, the low E string's resonant frequency is lower (or slower) than the thinner high E string. In this activity we'll observe how mass and stiffness affect the resonant frequency of different size rings. Get ready to shake things up!

Materials

• Scissors
• 4 sheets of construction paper (ideally four different colors)
• Tape
• Piece of cardboard (about five by 12 inches)
• Ruler

Preparation

• Cut seven lengthwise strips (about one inch wide) from the construction paper; cut two strips from the first three colors and one strip from the fourth.
• Use tape to connect the same colored strips, forming three long strips, each about 22 inches long.
• Keep one strip 22 inches long. Trim about three inches from the second strip and six inches from the third one. Combined with the strip cut from the fourth sheet, you should have four strips of paper with lengths of: 22, 19, 16 and 12 inches.
• Form the strips into rings by taping the two ends of each strip together.
• Tape the rings to the cardboard strip, leaving at least two inches between each strip

Procedure

• Place your ruler on a flat, clear surface.
• Place your cardboard sheet (with rings attached) on the same surface, perpendicular to your ruler, so that the short end of the cardboard is nearly touching the ruler. Line up one edge with the three-inch mark on the ruler.
• Gently move the cardboard about two inches along the length of the ruler, then move it back. Do this slowly a few more times. Notice the movement and shape of each paper ring as you move the cardboard. Are all the rings moving? Do some rings move more than others? Which ones move the most? Which ones move the least? What happens to the shapes of the rings as you move the cardboard?
• Repeat the movement but this time move the cardboard slightly faster. Again, pay attention to what the paper rings do as you move the cardboard. Do different rings move when you increase the speed? What happens to the shapes of the rings when you increase the speed of the cardboard? If more than one ring is moving, are they moving together (in synchrony)? Are any of the rings not moving? What happens to their shapes?
• Repeat the movement, slowly increasing the speed that you move the cardboard. Make sure to keep the movement to two inches. Every time you increase the speed of the movement, notice the effect on the rings. Notice whether the rings are moving and also whether their shapes change as you increase the speed. Keep increasing the speed to try to get all of the rings to move. Which ring was the last to move? Which ring was the first to move? What changed about the movement of the big ring as you increased the speed? What changed about the movement of the small ring as you increased the speed? What changed about the shapes of the rings as you increased the speed of the movement? Were you ever able to get all the rings to move back and forth at the same time?
• Try to find the resonant or “favorite” frequency for each ring. Increase and decrease the speed that you move the cardboard, watching to see the point where each ring seems the most excited, where that ring's movement is stronger and clearer than the other rings. Test to see if you can find a speed where only the smallest ring moves, then see if you can find the speed where only the biggest one moves. Test if you find a speed where all the rings move together. Which ring seems to move the most at lower speeds? Which one moves the most at higher speeds?
• Repeat the activity but this time try moving the cardboard six-inch increments back and forth. Pay close attention to what happens to the rings as you slowly increase the speed of the movement. Which ring moves first when you move the cardboard six inches? Is it the same ring that moved first when you moved the cardboard two inches? What happens to the shapes of the rings when you increase the distance of the movement? Is it easier or more difficult to get all the rings to move when you're moving the cardboard six inches back and forth?
• Repeat the activity, moving the cardboard back and forth by nine, then 12 inches. Pay attention to which rings move first and which rings move last at each distance. Also notice the shapes of the rings and how they change as you move the cardboard faster at each length. How do the sizes of each ring relate to the distance of movements?
• Extra: Repeat this activity but now hold the board above the table and move it up and down. Experiment with increasing the speed and distance that you are moving the cardboard. Notice how this affects the shape and movement of the rings.
• Extra: Repeat this activity using different materials to make the rings. Some materials you might try include aluminum foil, thin floral wire (be sure to ask for an adult's help!) and paper with different thicknesses. Notice how the stiffness of the material affects the rings’ movements and shapes.

Observations and results
Did you notice that at each distance, each ring seemed to have a favorite speed, a speed where that ring in particular seemed to have a stronger movement than the others? This is what we expect to see. The largest ring has the most mass, and it is also the floppiest (or least stiff). Just like with the low E guitar string, having more mass means the biggest ring has the lowest resonant frequency. Therefore, when you were moving the cardboard slowly, the biggest ring was probably more dynamic than the other rings. In contrast, the smallest ring has the smallest mass and is the least floppy (or the most stiff). As a result, the small ring has a higher resonant frequency and was the most dynamic when you were moving the cardboard faster.

If you tested different speeds of the movement, you might have noticed that at least some of the rings had more than one resonant frequency. For example, the big ring vibrated strongly when you were moving the cardboard slowly, but as you sped up it did not move as well. Then, when you got fast enough the big ring seemed to get going again! This is because the rings (like many objects) have multiple resonant frequencies. If you pay close attention, however, you will notice that the shape of the big ring is different at the low resonant frequency compared with the higher one. At the low frequency it flattens itself out whereas at the high one it might almost look like a square!

As you increased the distance of the movement, the resonant frequencies didn't change, but it was probably easier to see how the rings changed shape in response to the movement. If you moved the cardboard up and down, you probably noticed the rings followed this movement—instead of moving side to side they seemed to get skinny and fat. The biggest ring still has the lowest resonant frequency but you might have noticed that it was a little harder to get the smallest ring to move compared with when you moved the cardboard side to side. This is because when you're moving the board up and down, the cardboard is adding it's own stiffness to the rings, making them less floppy in that direction.

More to explore
What Material Makes the Most Resonant Soundboard?, from Science Buddies
The Coolest Things That Sound Waves Do, from DNews
Natural Frequency, from the Physics Classroom
Science Fair Project Ideas, from Science Buddies

This activity brought to you in partnership with Science Buddies