Two rulers and a pin
Tonight's idea
Hang a small weight on a string and let it swing a little. It keeps a steady beat. For small swings, the pull back towards the middle is close to proportional to how far you moved it: twice as far, twice the pull. Motion like that is regular. You can predict it far ahead.
Now pin two rulers end to end, so the second hangs from the first, and hang the pair from a nail. This is a double pendulum. Give it a small swing and it is still tame. Lift it high and let go, and the lower ruler whirls, stalls and flips. In big swings the pull is no longer proportional to the displacement, and the two rulers keep passing energy back and forth.
Chaos: the motion follows exact rules, yet a tiny difference at the start grows by multiplying, until two runs look nothing alike.
In a model with no friction, the total energy never changes, so the motion never settles down. Real rulers lose energy to friction and stop.
Watch one: the steady beat of a single pendulum
A small weight hangs on a string 1 m long. How long does one full swing, there and back, take?
For small swings of a small weight on a light string, the time \(T\) for one full swing (the period) is close to
\[T \approx 2\pi\sqrt{\frac{L}{g}},\]where \(L\) is the length in metres and \(g \approx 9.8\) measures the strength of gravity at the Earth's surface. With \(L = 1\):
\[T \approx 2 \times 3.1416 \times \sqrt{\frac{1}{9.8}} \approx 6.2832 \times 0.3194 \approx 2.0 \text{ seconds}.\]Notice what is missing from the formula: the size of the swing. Small swings of different sizes take almost the same time. The formula is an approximation: within 1% for swings up to about 20° each side of straight down, and worse for bigger swings.
Your turn
- Use the formula from the example, for small swings. (a) Find the period of a pendulum 0.25 m long. (b) Find the period of a pendulum 4 m long. (c) To double the period, how many times longer must the string be?
- A double pendulum made of two 30 cm rulers is released from high up, twice. The two starting positions of its lower tip are 1 mm apart. Suppose the gap between the two runs doubles every 0.5 seconds. (That is a made-up round figure. Real gaps grow unevenly.) (a) How big is the gap after 2 seconds? (b) After how many seconds is it first more than 50 cm? (c) The gap can never pass 120 cm. Why? What does that say about the doubling rule?
- Keep the doubling rule. Call a prediction "ruined" once the gap is more than 50 cm. (a) You start 1000 times more precisely: the tips begin 0.001 mm apart. After how many seconds is the prediction ruined? (b) What if they begin 0.000001 mm apart, a million times more precise than in exercise 2? (c) Each time the start is 1000 times more precise, how many extra seconds of prediction do you gain?
- Plan an experiment with two rulers, a pin and a nail. (a) How would you release the pendulum twice from nearly the same position and compare the two runs? (b) What do you expect for a small release, and for a high one? (c) Imagine no friction. You release both rulers from rest, pointing straight out sideways, level with the nail. Could both rulers later point straight up at the same moment? Use energy to explain.
Stretch: when twice as far is not twice the pull
Hold a pendulum weight out at an angle from straight down and gravity pulls it back. For one weight the pull, in some units, is 87 at 5°, 174 at 10°, 342 at 20°, 643 at 40°, 985 at 80° and 342 at 160°. Each angle is double the one before. How many times bigger does the pull get at each doubling? Where is "twice as far, twice the pull" a good rule, and where does it fail? A single pendulum swung high also breaks the rule, yet it is not chaotic. What does a double pendulum have that it lacks?
Answers
- (a) \(2\pi\sqrt{0.25 \div 9.8} \approx 6.2832 \times 0.1597 \approx 1.0\) second. (b) \(2\pi\sqrt{4 \div 9.8} \approx 6.2832 \times 0.6389 \approx 4.0\) seconds. (c) Four times longer. The period goes with the square root of the length, and \(\sqrt{4} = 2\). You can see it in the answers: 0.25 m, 1 m and 4 m give about 1, 2 and 4 seconds.
- (a) Two seconds is 4 doublings: \(1 \to 2 \to 4 \to 8 \to 16\) mm. (b) 50 cm is 500 mm. Eight doublings give \(2^8 = 256\) mm, which is not enough. Nine give \(2^9 = 512\) mm. Nine doublings take \(9 \times 0.5 = 4.5\) seconds. (c) The tip is never more than \(30 + 30 = 60\) cm from the nail, so two positions of the tip are at most 120 cm apart. So the doubling cannot go on for ever. The rule only describes the gap while it is still small. Once the gap is as big as the pendulum itself, the two runs are simply unrelated.
- (a) Now the gap must grow from 0.001 mm to more than 500 mm, a factor of more than 500,000. Since \(2^{18} = 262{,}144\) is too small and \(2^{19} = 524{,}288\) is enough, it takes 19 doublings: 9.5 seconds. (b) The factor is now more than 500,000,000. \(2^{28} = 268{,}435{,}456\) is too small and \(2^{29} = 536{,}870{,}912\) is enough: 29 doublings, 14.5 seconds. (c) Only 5 more seconds each time, because \(2^{10} = 1024\) is about 1000, so ten more doublings eat up the whole improvement. A million times the care bought 10 extra seconds. This is why chaotic motion cannot be predicted far ahead in practice, even though it follows exact rules.
- (a) One way: hold the tip of the lower ruler against a pencil mark on the wall behind, let go without pushing, and film it with a phone. Do it again from the same mark and play the two films side by side. (b) From a small release, the two runs should match swing after swing. From a high release, they should match for a second or two and then go separate ways. In a computer model of two 30 cm rulers with no friction, a 1 mm gap stayed under 2 mm for 12 seconds after a release at 10° from straight down, but passed 10 cm within 4 seconds after a release from level. Your hand cannot put the tip in exactly the same place twice, so there is always a small gap to grow. (c) No. At the start nothing is moving, and the middle of each ruler is level with the nail. With both rulers straight up, both middles would be above the nail. That needs more energy than the pendulum began with, and with no friction the total energy never changes. (The lower ruler alone can flip over the top, as long as the upper one is low enough to pay for it.)
Stretch. The pull grows by factors of \(174 \div 87 = 2.00\), then \(342 \div 174 \approx 1.97\), then \(643 \div 342 \approx 1.88\), then \(985 \div 643 \approx 1.53\), then \(342 \div 985 \approx 0.35\). Up to about 20° the rule is good. Past that it slips, and near the top it fails completely: going further gives a weaker pull. Up there a tiny difference can decide whether a ruler tips over the top or falls back, and that is how small gaps get multiplied. But breaking the rule is not enough for chaos. A single pendulum has only one moving part, and with no friction it just repeats the same swing. A double pendulum has two parts that push on each other and trade energy. Chaos needs both things at once. With small swings the rule nearly holds for both rulers, and the motion is close to regular: a blend of two steady beats.