Coupled Resonant Pendulums
Two pendulums suspended from a common support will swing back and forth in intriguing patterns if the support allows the motion of one pendulum to influence the motion of the other.
- A piece of string about two feet (60 centimeters) long
- PVC pipe and fittings to construct a suitable stand, or materials necessary to construct a stand of your own design—the stand shown above uses Schedule 40, 1/2-inch PVC pipe in the following lengths and their accessories:
- Two 12-inch pieces
- Three 3-inch pieces
- Two 90-degree elbows
- One T-joint
- One coupling
- Two jumbo paper clips
- Needlenose pliers
- Two 3/4-inch hex nuts (exact size is not critical)
- Optional: PVC cutter and hacksaw
- Assemble the PVC stand by fitting the pieces together as depicted in the photo above, or design your own stand. The purpose of the stand is to support the ends of the horizontal string, so anything that will accomplish this will suffice.
- Stretch the string between the two posts so that it is reasonably tight and secure the ends in place. Any way you can attach the string to the posts is okay—for example, you could cut slits in the top of the posts and insert the string, drill small holes and attach small screw eyes, or use duct tape.
- Use the needlenose pliers to bend the paper clips into hangers for the nuts as shown in the photos above. The two hangers should be as close to the same length as possible.
- Place the hangers on the string and put the nuts in place on the hangers as shown so that there are two pendulums hanging from the string approximately 2–3 inches (5–7 cm) apart.
Gently pull one pendulum back a short distance and let it go. As it swings back and forth, notice that the other pendulum also begins to move, picking up speed and amplitude with each swing.
Notice that the pendulum you originally moved slows down with each swing and eventually stops, leaving the second pendulum briefly swinging by itself. But then the process begins to reverse, and soon the first pendulum is swinging again while the second one is stopped.
The pendulums repeatedly transfer the motion back and forth between them this way as long as they continue to swing.
Experiment with different wire lengths, string tensions, string length, and masses to produce more strongly or weakly interdependent coupled pendulums.
Every pendulum has a natural or resonant frequency, which is the number of times it swings back and forth per second. The resonant frequency depends on the pendulum’s length. Longer pendulums have lower frequencies.
Every time the first pendulum swings, it pulls on the connecting string and gives the second pendulum a small tug.
Since the two pendulums have the same length, the pulls of the first pendulum on the second occur exactly at the natural frequency of the second pendulum, so it (the second pendulum) begins to swing too. However, the second pendulum will swing slightly out of phase with the first one. When the first pendulum is at the height of its swing, the second pendulum is still somewhere in the middle of its swing.
As soon as the second pendulum starts to swing, it starts pulling back on the first pendulum. These pulls are timed so that the first pendulum slows down.
To picture this, it may help you to think of a playground swing. When you push on the swing at just the right moments, it goes higher and higher. When you push the swing at just the wrong moments, it slows down and stops. The second pendulum pulls on the first pendulum at just the “wrong” moments.
Eventually, the first pendulum is brought to rest; it has transferred all of its energy to the second pendulum. But now the original situation is exactly reversed, and the first pendulum is in a position to begin stealing energy back from the second. And so it goes, the energy repeatedly switching back and forth until friction and air resistance finally steal all of it away from both pendulums.
If the two pendulums are not the same length, then the tugs from the first pendulum’s swings will not occur at the natural frequency of the second one. The two pendulums swing, but with an uneven, jerky motion.
It’s easy to predict how often the two swinging cans will trade energy. Count the total number of swings per minute when you start both pendulums together and they swing back and forth, side by side.
Compare that to the number of swings per minute when you start them opposite one another—that is, with one pulled forward and one pulled backward an equal distance from the string, and then released at the same time.
The difference between those two numbers exactly equals the number of times per minute that the pendulums pass the energy back and forth if you start just one pendulum while the other hangs at rest.
Physicists call these two particular motions normal modes of the two-pendulum system, and they call the difference between the frequencies of the normal modes a beat frequency.
