Coupled Resonant Pendulums

These pendulums trade swings back and forth.

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.

**2 plastic 35 mm film cans.****Clay, coins, or washers**for mass.**2 pieces of metal coat hanger wire,**each about 8 inches (20 cm) long.**A piece of string**about 3 feet (90 cm) long.**2 ring stands**or other vertical supports for the string.

(30 minutes or less)

Stretch and secure the string between two ring stands placed about 20 to 30 inches (50 to 75 cm) apart. In the center of each film can lid, punch a hole just large enough to insert one end of a coat hanger wire. Bend the end of the inserted wire so the lid won't slide off but so that you can still put the lid on the can. Bend the other end of the wire so it will hang freely from the string. The two hangers should be close to the same length. Add equal amounts of clay, coins, or washers to each can and attach the lids. Hang the pendulums so that they are about equally spaced from each other and from the ring stands.

(15 minutes or more)

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 and with different string tensions to produce more strongly or weakly interdependent coupled pendulums.

Every pendulum has a *natural* or *resonant frequency*, which is the number of times the pendulum 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 the second pendulum begins to swing too. The second pendulum swings slightly out of phase with the first one. That is, 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 is 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*.