## Pendulum Waves

posted on 2 Mar 2014 by guy

last changed 7 Jan 2016

Your vote (click to rate)ages: 12 to 99 yrs budget: $1.00 to $5.00prep time: 30 to 240 min class time: 10 to 240 min |
This lesson outlines the construction and operation of a pendulum wave machine that can be built from inexpensive materials in the classroom. Building the demonstration requires a fair amount of patience: construction only takes about 30 minutes, but adjustments and fine tuning can take many hours. The device beautifully illustrates the nature of pendulum oscillations. The lesson also includes a general discussion of pendulum timing, and a history of pendulums in clocks. The device is simple enough so that students with enough time can benefit from building, timing and fine tuning the device. The attached spreadsheet is designed to help with timing measurements and adjustments. |

**required equipment:**stop watch, yard stick, screw eyes, string

**subjects:**Engineering, Mathematics, Physics

**keywords:**simple, pendulum, period, wave

The pendulum demonstration in the video above uses carefully measured pendulums to achieve a delicate and magical synchronism. The pendulums start in synch, but as the demonstration proceeds, the longer pendulums lag behind the shorter pendulums. Somewhat surprisingly, after one minute all the pendulums come back into synch.

To achieve this precise choreography, the lengths of the pendulums are constructed such that over the course of one minute, the longest pendulum makes 51 oscillations, the next longest makes 52 oscillations, with each subsequent pendulum making one more oscillation than the last, all the way to the final pendulum (the 11^{th}), which makes 61 oscillations. After one minute, all of the pendulums have made an integer number of oscillations and are back where they started.

The device is constructed after a design by Richard Berg at the University of Maryland, published in 1991.1

## the simple pendulum

The pendulum wave is easy to understand in the case of the "simple pendulum", which has a fixed relation between the length of a pendulum and its period (the time it takes to swing back and forth).

The "simple pendulum" is an idealization of a real pendulum, and makes several assumptions about the construction and motion of the pendulum:

- The mass of the pendulum is concentrated in a small dense bob at the end of the string; the shape of the bob and the mass of the support string are irrelevant.
- Friction at the pivot point, buoyancy due to air pressure, and air drag on the bob and string are all negligible.
- The bob moves in a single plane, back and forth. It does not trace an ellipse, as in the conical pendulum.

Fig. 1: Comparison of the simple pendulum and a true pendulum for a 10^{o} amplitude. The true pendulum is shown as a solid bob, the simple pendulum is shown as an empty bob (indistinguishable in this animation). Image made available by Lookang under the CC-BY-SA-3.0 license at Wikimedia Commons.

Fig. 2: Comparison of the simple pendulum and a true pendulum for a 90^{o} amplitude. The true pendulum is shown as a solid bob, the simple pendulum is shown as an empty bob. The true pendulum lags behind the ideal simple pendulum. Image made available by Lookang under the CC-BY-SA-3.0 license at Wikimedia Commons.

length (cm) | periods/min. |
---|---|

34.4 | 51 |

33.1 | 52 |

31.8 | 53 |

30.7 | 54 |

29.6 | 55 |

28.5 | 56 |

27.5 | 57 |

26.6 | 58 |

25.7 | 59 |

24.8 | 60 |

24.0 | 61 |

Fig. 4: The homemade pendulum wave machine. Pendulum bobs are attached to screw eyes every 3 inches along a yard stick.

Under these conditions, the period of the pendulum is independent of the mass of the bob, which means that all simple pendulums oscillate the same way, no matter what you hang off the end of the string. This fact follows easily from Newton's laws of physics. As long as friction, buoyancy and drag are negligible, the only forces on the pendulum come from gravity and string tension. Both of these forces are proportional to the mass of the bob. Following Newton's second law ($a = {F\over m}$), the acceleration of the bob is given by the net force divided by the mass. In that calculation, the mass terms cancel so that the acceleration, and any motion caused by that acceleration, does not depend on the mass.

For small oscillations, the period is approximately independent of how far the pendulum swings back and forth (the "amplitude" of the swing). This fact may seem counterintuitive; it may seem a pendulum bob that travels farther should take longer to do so. However, as we raise the pendulum to higher amplitudes, the pendulum bob accelerates more quickly as it falls. This effect tends to offset the longer distance the bob travels in such a way that the total period is approximately independent of the amplitude (as long as the amplitude is not too big).

For simple pendulums adhering to the conditions stated above, the period of the pendulum depends only on the length of the string and the strength of gravity: $$T = 2\pi\sqrt{l\over g}$$ where $T$ is the period, $l$ is the length of the string (measured from the pivot point to the center of mass of the bob), and $g$ is the acceleration due to gravity (approximately 9.8 m/s^{2} at the surface of the earth). (See below for the derivation.) This formula is accurate to better than 1% for amplitudes less than 23^{o}, but for larger amplitudes the true period is longer. Figures 1 and 2 compare the motion described by the simple pendulum and the motion of a real pendulum for amplitudes of 10^{o} and 90^{o}. The solid pendulum bob shows the true motion, while the hollow bob shows the simple pendulum motion with the period given above. In figure 1, for a 10^{o} amplitude, the two motions overlap and are indistinguishable. In figure 2, for a 90^{o} amplitude, the true motion lags behind the ideal calculation.

## pendulums as clocks

Galileo was the first person to publish studies of the timing of pendulums. He first described his research in a letter to Guido Ubaldo, dated November 29, 1602. According to his student, Vincenzo Viviani, Galileo's interest in pendulums was initially stimulated while watching a swinging chandelier in the Pisa cathedral. Apparently, the oration that day was not very engaging.

Galileo determined, somewhat to his surprise, that the period of oscillation did not depend appreciably on the size of the amplitude, a discovery that suggested a pendulum could make a reliable clock. (He discovered this fact by comparing the swing of a pendulum with the timing of his own pulse.) In 1656, the Dutch scientist Christiaan Huygens built the first pendulum clock based on that idea. By 1670, the recognition that only small oscillations were independent of amplitude led to the invention of the anchor escapement, which limited the swing of the clock pendulum to about 6^{o}. The grandfather clock, invented by the English clock maker William Clement, used this technology with a 1-meter-long pendulum, for which each swing (half period) took one second.

Around the same time, it was also recognized that the period of the pendulum could vary with the acceleration due to gravity. In 1671, Jean Richer observed that a pendulum clock at Cayenne, French Guiana was 2.5 minutes slower per day than a similar clock at Paris. This observation led him to conclude (correctly) that the acceleration due to gravity was lower at Cayenne. In general, the (apparent) acceleration due to gravity varies with latitude because of the oblateness of the earth and because of centripetal effects due to the earth's rotation. It also varies with altitude.

## build it yourself

The device in the video is constructed from common household items. To build it, you'll need the following materials:

- wooden meter stick (or a yard stick for you Yanks),
- 11 small screw eyes,
- 11 binder clips,
- a thumbtack,
- thread or light string,
- and a collection of small heavy items to use as pendulum bobs.

Since the mass and shape of the bob doesn't matter much (as long as it's small and heavy) we can use anything that we can tie a string onto: keys, fishing weights, nuts and bolts, jewelry, rocks, washers, toy soldiers, flashlights, cell phones, you name it. The best items are small, symmetrical and heavy, with enough inertia so that they are not slowed appreciably by friction or air drag.

Put small screw eyes (about 3/16 inch thread length) into the yard stick every 3 inches (figure 4). You can use the thumb tack to start each hole before screwing the eye in. Collect a number of small items you wish to use as pendulum bobs and tie some light string onto them. Attach each bob onto a screw eye, using different length strings for each bob. Choose each length according to the period you want. The pendulum lengths used in the video are given in table 1, if you wish to reproduce that system.

Wrap the string tightly around the screw eye until you get the right length. Hold the string in place with a binder clip (see figure 5).

Time each pendulum carefully with a stopwatch and, if necessary, adjust the length of the string by turning the screw eye appropriately. Timing and length adjustment is the hardest part of the construction. See under "troubleshooting" below for some problems to watch out for. Use the attached spreadsheet to help with timing measurements and calculations if you wish.

Place the ends of the meter stick on two chairs, spanning the space between the chairs. Weigh the stick down with books or tape it in place. Pull all the bobs back together using a board or book (as in the video) and release.

Enjoy the show.

## period derivation (for geeks who know calculus)

The differential equation describing the motion of a pendulum derives from Newton's second law. In the direction perpendicular to the string, the force on the bob comes only from the component of the gravity force in that direction. Using the parameters labeled in figure 3, with the force in the negative $\theta$ direction, the second law gives: $$F=-mg\sin\theta=ma=ml{d^2\theta\over dt^2}$$

which rearranges to: $${d^2\theta\over dt^2}+{g\over l}\sin\theta=0$$

This equation does not have a solution in terms of elementary functions, but in the case where the angular displacement $\theta$ is small, $\sin\theta$ can be approximated by $\theta$ (in radians) and the equation reduces to: $${d^2\theta\over dt^2}+{g\over l}\theta=0$$ for which the solution is: $$\theta(t)=\theta_0\cos\left(\sqrt{g\over l}t\right)$$

This solution describes an oscillating system with a period of $T=2\pi\sqrt{l\over g}$.

The solution for arbitrary-size oscillations is more complicated. See the wikipedia article at http://en.wikipedia.org/wiki/Pendulum_%28mathematics%29 for details.

## question to ponder

**What would happen to a pendulum if we took it to the moon?**

An expensive but interesting experiment! On the moon, the surface gravity is 1.62 m/s^{2}, about ^{1}/_{6} of what it is on earth. As a result, the gravitational force is weaker there and objects in free fall accelerate more slowly. The pendulum moves more slowly also. Its period increases by a factor of $\sqrt6=2.45$.

## troubleshooting

Getting the timing right is the hardest part. Time each pendulum bob individually with a stop watch to make sure the period is what you expect. To get reasonable synchronization, the periods need to be accurate to about one part in 2000.

To verify that each bob has the correct period, time a fixed number of oscillations. Even with a good stopwatch, most people can only start and stop the clock to an accuracy of about 0.1 seconds. I found I was able to reliably measure to an accuracy of about 0.06 seconds (RMS) using the stop watch app on my iPhone, but only after several days of practice. To achieve the necessary accuracy, you'll need to time oscillations for 2 to 3 minutes. If the first bob is supposed to make 51 oscillations in one minute, time 153 oscillations and adjust it as close to 3 minutes as possible. Make at least three separate measurements of each pendulum bob to make sure you are getting consistent measurements. Use the attached spreadsheet to help record your measurements.

While making timing measurements, be sure that the amplitude of the oscillations is small, 5 degrees or less. A larger amplitude will slow the oscillation appreciably. At 23 degrees, the oscillation is 1% slower, much more than can be tolerated for good synchronization.

For a large bob, the moment of inertia of the bob about its own center of mass may measurably alter the moment of inertia of the pendulum as a whole. Make sure you swing large irregular objects always the same way. The period can vary depending on which way the bob is facing. Generally, small compact bobs are easier to work with.

If all else fails, you can use longer string lengths to slow the oscillation and make the synchronization less sensitive to timing errors. Quadrupling the lengths will double the periods.

## further resources

There is a rich literature on both pendulums and pendulum clocks. Here are just a few sources to get you started if you want to go further.

- Wikipedia posts a very careful and thorough discussion of a huge range of topics pertaining to pendulums.
- Albert Van Helden Albert at
*The Galileo Project*has posted an erudite description of the history of the "Pendulum Clock". - A detailed but somewhat technical discussion of the differences between the simple pendulum and real life pendulums is presented in Robert Nelson and M. G. Olsson. "The pendulum – Rich physics from a simple system".
*American Journal of Physics***54**(2) Feb 1987: 112–121. - The "Horologium Oscillatorium" by Christian Huygens is translated by Ian Bruce at 17thcenturymaths.com.
- The folks at hyperphysics provide calculators for small amplitude periods for both the simple pendulum and pendulums with arbitrary bobs (if you know the moment of inertia).

- 1. Richard E. Berg. "Pendulum waves: A demonstration of wave motion using pendula" American Journal of Physics 59 1991: 186.

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