# Pendulum Snake

Ten pendulums swing back and forth at slightly different rates, determined by the length of the strings. Starting in sync, the pattern shifts over time, until finally the pendulums realign.

- PVC pipe, schedule 40, 1/2 inch (1.25 cm) inner diameter, 10 feet (3 m) long

Note: This is the standard length you’ll find in home-improvement stores. For ease of transport, you can have it cut into two 5-foot (1.5-m) pieces. - PVC cutters
- Measuring tape or meter stick with both English and metric units
- Straightedge at least 2 feet (60 cm) long
- Pencil
- Drill press or drill guide
- Drilling jig (or wood blocks, hot glue, and clamps to construct one), or other method to secure the PVC while drilling
- Electric drill and 1/8-inch (3 mm) bit
- Twenty Phillips pan-head sheet-metal screws, #6 x 1/2 inch (13 mm)
- Phillips-head screwdriver
- 30-foot (9-m) length of string (braided string, such as mason twine or chalk twine, will work best, since they won't unravel)
- Scissors
- Ten 3/4-inch (2-cm) hex nuts
- Tapestry needle, #16, straight (not curved), available at craft stores (if unavailable, a paper clip may be substituted; instructions are below)
- Two 1/2-inch (1 cm) PVC 90-degree elbows
- Two 1/2-inch (1 cm) PVC T-joints
- Four 1/2-inch (1 cm) PVC couplings or caps
- 21-inch (50-cm) length of 1 x 4 wood, or equivalent

Note: “1 x 4” is the standard designation for lumber that is actually 3/4 inches (2 cm) thick and 3 1/2 inches (9 cm) wide.

Note that this Snack is constructed in four parts. In Part I, you’ll cut the PVC and mark the locations of the holes from which the pendulums will hang. In Part II you’ll drill holes in the pipe. In Part III, you’ll add the pendulums. In Part IV, you’ll finalize assembly and set the Pendulum Snake in motion.

*Part I: Cut the PVC and Mark the Locations of Holes*

- Using the PVC cutters and the measuring tape with metric units, cut the following seven lengths of PVC pipe:
- One 58-cm piece
- Two 45-cm pieces
- Four 25-cm pieces

Set aside all but the longest length of pipe. The other pieces will be used to construct the frame in Part IV.

- Using a pencil and straightedge, draw three straight lines along the length of the 58-cm length of pipe. The lines should be parallel to each other and approximately 1 cm apart.
- Working only on the center line, set the pipe horizontally and begin making these marks:
- Make the first mark 5 cm from the end of the pipe
- Make a second mark 3 cm farther on
- Make a third mark 2 cm from the second mark
- Make a fourth mark 3 cm farther on
- Make a fifth mark 2 cm from that mark

- With your pencil, number the dots you made, from 1 to 20.
- Move to the top line and make a dot directly above each odd number marked on the center line (that is, above the 1, 3, 5, and so on). Then move to the bottom line and make a dot directly below each even number marked on the center line (that is, below the 2, 4, 6, and so on). When you’re done, you should have 10 marks on the top line, 20 (numbered) marks on the center line, and 10 marks on the bottom line (click to enlarge the diagram below).

*Part II: Drill the Holes*

- Find a way to hold the marked-up PVC pipe firmly in place so you can drill into it. Orient the pipe so that the center line, with its penciled-on numbers, runs along the top of the pipe.
Note: A drilling jig will hold the pipe steady, keeping it from rolling while you work and protecting the surface beneath it. If you don’t have a drilling jig, you can make one by hot-gluing scrap blocks of wood to a base, and then clamping the jig to a table or drill press (see photo below).

- Once the pipe has been secured, it’s time to begin drilling. The first holes will be made on the numbered marks of the center line. However, you’ll need to make holes that go through both the top wall of the pipe and the bottom wall, directly under them.
You can either drill both sets of holes at once (see Option A below), or you can drill one side, turn the pipe, and drill the other (see Option B below). Use Option A if you have access to a drill press and are familiar with its use, or if you are confident in your ability to drill almost perfectly vertically with a hand-held drill. (An inexpensive drill guide may help.) Otherwise, use Option B.

- Option A:
At each mark on the middle line, drill 1/8-inch (3 mm) holes through the top wall of the pipe and continue drilling through the bottom wall.

- Option B:
At each mark on the middle line, drill 1/8-inch (3 mm) holes through the top wall of the pipe. When you’re done, rotate the pipe so the line of holes you drilled sits on the bottom of the pipe. Then, on the top of the pipe (directly above the set of holes you just drilled) draw a straight line the length of the pipe and mark it in exactly the same way you did in Step 3 above. When you’re done, drill 1/8-inch (3 mm) holes at those marks, as you did on the opposite side of the pipe.

- Option A:
- When the center-line holes have been drilled, reorient the pipe so one of the two “side lines” is now at the top of the pipe. At each of the marks on the line, drill a 1/8-inch (3 mm) hole through the first wall of the pipe. (Do not drill all the way through the opposite wall!) You should have 10 holes when you’re done.
- Turn the pipe again and repeat for the second side line. Once all the holes have been drilled, you’re ready to add the pendulums.
- Screw a sheet-metal screw several threads (not all the way) into each hole that you’ve drilled on the two side lines. There should be a total of 20 screws, 10 on each side line (click to enlarge the photo below).

*Part III: Add the Pendulums*

- With your scissors, cut 10 pieces of braided string as follows:
- Two 100-cm pieces
- Two 90-cm pieces
- Two 80-cm pieces
- Four 70-cm pieces

- Thread a 100-cm piece of string onto the tapestry needle. Thread the needle down through the first set of holes (top to bottom) on the center line. Thread a hex nut onto the string, and then thread the string back up through the next set of holes (bottom to top) on the center line (see photo below).
Note: If you don’t have a tapestry needle, straighten out a paper clip and use a small piece of masking tape to hold the wire and string together.

- Remove the tapestry needle and wrap each end of the string around the screw nearest it on the side line. Then tighten the screws enough to hold the string in place (see photo below).
- Repeat Steps 2 and 3 for the second 100-cm piece of string, using the third and fourth sets of holes. (Don't forget to add the hex nut!) Then use the same process to add the rest of the pendulums, as follows:
- Repeat for the two 90-cm pieces of string, using the 5th and 6th sets of holes and the 7th and 8th sets of holes.
- Repeat for the two 80-cm pieces of string, using the 9th and 10th sets of holes and the 11th and 12th sets of holes.
- Repeat for the four 70-cm pieces of string, using the four remaining pairs of sets of holes (13 and 14; 15 and 16; 17 and 18; 19 and 20).

- You should now have 10 loops with hex nuts on them: Each is an individual pendulum.

*Part IV: Assemble the Frame and Set the Pendulum Snake in Motion*

- Use the remaining pieces of PVC to assemble the frame as shown in the photo below (click to enlarge). Once the assembly is complete, it’s time to adjust the lengths of the pendulums.
- Starting with the first pendulum you threaded (the first 100-cm length of string), carefully adjust each to the following lengths: 38.8 cm, 35.8 cm, 33.1 cm, 30.7 cm, 28.5 cm, 26.6 cm, 24.8 cm, 23.3 cm, 21.8 cm, 20.5 cm
To do this, measure from the bottom of the PVC pipe, midway between the two hanging strings, to the center of the hole in the hex nut (the hex nut’s center of mass). When the pendulum is the correct length, firmly tighten the screws, leaving any extra string in place.

- Using the same method, adjust all the strings until the desired length is achieved and all the screws have been tightened.
- Finally, double-check the length of each pendulum. Then use your scissors to clip off any excess string that might get in the way of its movement. (Be sure to leave a bit of string, though, in case you need to make later adjustments.) Set the 21-inch (50-cm) board nearby to get the Pendulum Snake in motion, and you’re ready to go (see photo below).

Face the row of pendulums and use the wooden board to pull them all towards you simultaneously (see photo below). Flip down the board to get it quickly out of the way, and then sit back and watch the show!

At first, all the pendulums move together. Then a snake-like pattern emerges (see left photo below), dissolving into a random mess. Next, the pendulums alternate and move in perfect opposition (see right photo below). The random mess reappears, eventually followed by a return to the original perfect alignment. Then the whole cycle repeats.

If you don’t see these patterns, try “tuning” the pendulums by adjusting their lengths slightly. If a particular pendulum is arriving too soon at the moment when everything should be synced, then lengthen it a little; if it’s arriving late, then shorten it a little. Adjust the string length by loosening either screw, shifting the string, and then retightening the screw.

Each pendulum swings back and forth at a slightly different rate, determined by the length of string supporting it. This causes them to fall out of step with each other as they swing.

At certain time intervals, however, the balls fall into recognizable patterns. For example, after 15 seconds, neighboring pendulums are one half-swing apart, causing them to swing in perfect opposition. After 30 seconds, the balls line up again and the cycle of patterns repeats (see table below).

Length (cm) | Number of Back and Forth Swings in 30 seconds |
---|---|

38.8 | 24 |

35.8 | 25 |

33.1 | 26 |

30.7 | 27 |

28.5 | 28 |

26.6 | 29 |

24.8 | 30 |

23.3 | 31 |

21.8 | 32 |

20.5 | 33 |

People are often surprised to learn that the time it takes a pendulum to swing back and forth—called the pendulum’s period—doesn’t depend on the mass of the pendulum bob or on how far the pendulum swings.

You can prove this to yourself by swinging a bottle of water on a string: First try it with a full bottle, then dump out half of the water. Then try little swings and big, wide swings. Does the pendulum’s period change?

**Math Root**

The equation for the period (T) of a simple pendulum is:

$$T=2 \pi \sqrt{\frac{L}{g}}$$

where *T* is the period in seconds per swing, *L* is the length (in meters) and *g* is the acceleration due to gravity (9.8 m/s^{2}).

The frequency *F* of a pendulum is the number of times it swings all the way back and forth in one second (expressed in swings per second). Frequency is the inverse of period, or 1/*T*. So the equation for calculating the frequency of a pendulum is the inverse of the formula above, or

$$F=\frac{1}{2 \pi \sqrt{\frac{L}{g}}}$$

Squaring both sides,

$$F^2=\frac{1}{4 \pi^2 \frac{L}{g}}=\frac{g}{4 \pi^2 L}$$

Solving for L,

$$L=\frac{g}{4 \pi^2 F^2}$$

You can use this equation to calculate the length of a pendulum for a given frequency. For example, the longest pendulum of your Pendulum Snake swings back and forth 24 times in 30 seconds. Frequency is expressed in swings per second, so 24 swings/30 seconds = 0.8 swings/second. Using 9.8 m/s^{2} for *g* and 0.8 swings/second for *F* in the equation gives you a length of 0.387 meters, or 38.7 centimeters.

Check out this episode of Build Your Own Exploratorium to see this Snack in action.