a × S = √2V
where a is the radius of the boomerang—the distance from its center to its tip. By flicking your wrist more, you can increase S in an attempt to get the formula to work.
Figure 5.4 The top of the boomerang, A, travels faster than the bottom, B, thanks to the spin.
The angle at which the boomerang tilts depends on the difference in the forward speed of the top and bottom of the boomerang. The top is traveling at speed V + aS while the bottom is going slower, at speed V – aS, where S is the angular speed measuring the rate at which the boomerang is spinning around its center (see figure 5.5). You can therefore vary the tilt of the boomerang by changing the speeds V and S, which will have an effect on how quickly the boomerang precesses, or twists, as it moves around its circular arc at speed V. If your boomerang won’t come back, it may be that you aren’t flicking it at the right speed S in relation to the launch speed V. This equation will help you adjust your throw accordingly.
Once you’ve mastered how to get the boomerang to come back to you, will throwing it harder and faster make it turn in an even bigger arc? The mathematics can be pieced together to give an equation that tells us the radius of the boomerang’s circular trajectory. Once again, the equation is a recipe that takes various ingredients that define the boomerang and its flight, mixes them together, and outputs the radius. Here are the ingredients:
• J, the moment of inertia of the boomerang. This is a measure of how hard it is to spin the boomerang; the heavier the boomerang, the greater J is. The moment of inertia also depends on the boomerang’s shape.
• ρ, the density of the air through which the boomerang flies
• C L, the lift coefficient—a number that determines the amount of lift the boomerang experiences, depending on its shape
• π, the number 3.14159 . . .
• a, the radius of the boomerang
The radius of the boomerang’s path R is determined by mixing these ingredients together using the following recipe:
With this equation, we see that if we throw the boomerang harder and faster, it doesn’t change the radius because speed isn’t one of the ingredients in the recipe. But what happens if we make the boomerang heavier by sticking a bit of Blu-Tack at the end of each wing? The equation helps us to predict that an increase in weight will increase the moment of inertia J, and that will increase the radius R. So a heavier boomerang goes around in a bigger circle—useful to know before throwing boomerangs around in a confined space!
You can download a PDF from the Number Mysteries website with instructions to make your own boomerang.
Can You Make an Egg Defy Gravity?
Take a hard-boiled egg. Lay it on its side on a table and give it a spin. Miraculously, the egg stands up, seemingly defying the laws of gravity. Even more strangely, if you try it with a raw egg, the same magic doesn’t happen.
It took until 2002 for mathematicians to come up with an explanation for this behavior. The rotational energy is translated, via the friction on the table, into potential energy, which pushes the egg’s center of gravity upward. If the table has no friction or too much friction, then the effect doesn’t happen. Part of the energy transferred to a raw egg gets absorbed by the fluid interior, and there isn’t enough left to push the egg upright.
WHY ARE PENDULUMS NOT AS PREDICTABLE AS THEY FIRST APPEAR?
It was Galileo, the master of using math to make predictions, who first unlocked the secret of what makes a pendulum tick. The story goes that when he was 17, he was attending Mass at the cathedral in Pisa. In a moment of boredom, he stared up at the ceiling, and his eyes fell on a chandelier that was swinging gently in the breeze blowing through the building.
Galileo decided to time how long it took the chandelier to swing from side to side. He didn’t have a watch (they hadn’t been invented yet), so he used his pulse to keep track of the swing. The great discovery he made was that the time the chandelier took to complete one swing did not seem to depend on the size of the swing. In other words, the time of the swing essentially doesn’t change if you increase or decrease the angle of swing. (I put the word essentially in there to indicate that if we dig a little deeper, things get slightly more complicated.) When the wind blew harder, the chandelier swung through a larger arc but took the same time to swing as when the wind dropped and the chandelier was hardly moving at all.
This was an important discovery and resulted in the swinging pendulum being used to record the passage of time. If you are starting a pendulum clock, you don’t have to worry about how far to the side you are going to move the pendulum, especially as the angle of swing will decrease over time. But what does the time of the swing depend on, and can we predict whether and how the swing will change if the weight is increased or if the pendulum is made longer?
As we might guess from Galileo’s experiment at the Leaning Tower of Pisa, a heavier pendulum does not travel faster, so a pendulum’s swing does not depend on its weight. But increasing the length of the pendulum does have an effect on the time of the swing. It turns out that multiplying the length by 4 doubles the time. Multiply the length by 9 and the time triples; multiply it by 16 and the time quadruples.
Again, an equation can capture this prediction. The time of swing T goes up in step with the square root of the length L:
This is actually another way of writing the equation that Galileo created for dropping balls from the Leaning Tower: the g is, again, the acceleration due to gravity. The reason for the ≈ as opposed to an = and my earlier use of essentially is that this is a good approximation for the time a pendulum takes to swing from one side to the other. As long as the swing isn’t too big, it’s possible to use this equation to predict the pendulum’s behavior. But if the angle of swing is large—if we start the pendulum almost vertically, for example—then the math gets much more difficult. Now the angle starts to have an effect on the time of swing, which Galileo didn’t pick up because the chandelier in the cathedral couldn’t swing that far. We also don’t see the effect in a grandfather clock, because the pendulum’s swing is quite small.
The mathematics you need to find the right equation to predict the behavior of a pendulum with a large angle of swing goes beyond what is taught in most math degree courses. Here is the beginning of the formula. It actually has an infinite number of parts that contribute to the behavior of the pendulum. θ0 is the initial angle the pendulum makes with a vertical line.
But this is nothing compared with the problem of predicting the behavior of a slightly modified pendulum. Instead of a single rigid rod swinging to and fro, imagine adding a second pendulum hinged to the bottom of the first, so that the whole thing looks a little like a leg, with an upper and lower part hinged at the knee. Predicting the behavior of this double pendulum is extremely complex. It’s not that the equations are that much more complicated, but that their solutions are very unpredictable: by changing the initial position of the pendulum very slightly, the result can be dramatically different. This is because the double pendulum is an example of a mathematical phenomenon called chaos. But a double pendulum isn’t just an amusing desktop game. The math behind it has important consequences for a question that could affect the future of humanity itself.
One of the many computer simulations of the double pendulums found online can be viewed at www.myphysicslab.com/dbl_pendulum.html. You can also watch it by using your smartphone to scan this code.
Try to predict whether the lower piece will next pass clockwise or counterclockwise through the top pendulum. It is almost impossible.
To make your own pendulum, see www.instructables.com/id/The-Chaos-Machine-Double-Pendulum or scan this code with your smartphone.
WILL THE SOLAR SYSTEM FLY APART?
Since Galileo first investigated falling balls and swinging pendulums, mathematicians have formulated hundreds of thousands of equations that predict how nature behaves. These equations are the foundations of modern science and are known as laws of nature. Math has en
abled us to create the complex technological world in which we live. Engineers rely on equations to reassure them that bridges won’t fall down and airplanes will stay in the air. You might think from our story so far that predicting the future would always be easy, but it’s not always that simple—as the French mathematician Henri Poincaré discovered.
In 1885, King Oscar II of Sweden and Norway offered a prize of twenty-five hundred kroner for anyone who could establish mathematically once and for all whether the solar system would continue turning like clockwork, or whether it was possible that at some point, the earth might spiral away from the sun and off into space. Poincaré thought he could find the answer, and he began to investigate.
One of the classic moves that mathematicians make when they are analyzing complicated problems is to simplify the setup in the hope that it will make the problem easier to solve. Instead of starting with all the planets in the solar system, Poincaré began by considering a system with just two bodies. Isaac Newton had already proved that their orbits would be stable: the two bodies just travel in elliptical orbits around each other, forever repeating the same pattern.
Figure 5.5
From this starting point, Poincaré began to explore what happens when another planet is added into the equation. The problem is that as soon as you have three bodies in a system, for example the earth, moon, and sun, the question of whether their orbits are stable gets very complicated—so much so that it had stumped even the great Newton. The problem is that now there are 18 different ingredients to combine in the recipe: the exact coordinates of each body in each of three dimensions, and their velocities in each dimension. Newton himself wrote that “to consider simultaneously all these causes of motion and to define these motions by exact laws admitting of easy calculation exceeds, if I am not mistaken, the force of any human mind.”
Poincaré was not daunted. He made significant headway, simplifying the problem by making successive approximations to the orbits. He believed that if he rounded up or down the very small differences he found in the positions of the planets, it wouldn’t affect his final answer too much. Although he couldn’t solve the problem in its entirety, his ideas were sophisticated enough to win him King Oscar’s prize. However, when Poincaré’s paper was being prepared for publication, one of the editors couldn’t follow Poincaré’s mathematics and raised a question. Could Poincaré justify why making a small change in the positions of the planets would result in only a small change in their predicted orbits?
While Poincaré was trying to justify his assumption, he suddenly realized he’d made a mistake. Contrary to what he’d originally thought, even a small change in the initial conditions—the starting positions and velocities of the three bodies—could end up producing vastly different orbits: his simplification didn’t work. He contacted the editors and tried to stop the paper from being printed, because publishing an erroneous paper in honor of the king would have caused a furor. The paper had already been printed, but most of the copies were gathered together and destroyed.
It all seemed a huge embarrassment. But as often happens in mathematics, when something goes wrong, the reason it goes wrong leads to interesting discoveries. Poincaré wrote a second, extended paper explaining his belief that very small changes could cause an apparently stable system to suddenly fly apart. What he discovered through his mistake led to one of the most important mathematical concepts of the last century: chaos theory.
Poincaré had found that even in Newton’s clockwork universe, simple equations can produce extraordinarily complex results. This isn’t the mathematics of randomness or probability. We are dealing here with a system that mathematicians call deterministic: it’s controlled by strict mathematical equations, and for any set of starting conditions, the outcome will always be the same. A chaotic system is still deterministic, but a very small change in the starting conditions can lead to a vastly different outcome.
Here is a small-scale example that acts as a good model of the solar system. We put three magnets—one black, one gray, and one white—on the floor. Above the magnets, we set up a magnetic pendulum that is free to swing in any direction. The pendulum will be attracted by all three magnets and will swing among them before assuming a stable position. At the end of the pendulum is a cartridge that drips out a trail of paint. We set the pendulum swinging, and the dripping paint traces out the path of the pendulum. What we are really attempting to simulate is an asteroid whizzing through a solar system with three planets attracting it: eventually, the asteroid will hit one of the planets.
The extraordinary thing is that it is almost impossible to repeat the experiment and get the same paint trail. As hard as you try to set the pendulum off from the same position and in the same direction, you find that the paint traces out a completely different path and ends up attracted to a different magnet each time. Here are three paths that start in almost the same way but end up at different magnets:
Figure 5.6 Just a small change in the initial position of the pendulum can cause it to follow a completely different path between the three magnets (shown as the small white, gray, and black dots).
The equations controlling the path of the pendulum are chaotic, and just a very small change in the starting location has a dramatic effect on the outcome. This is the signature of chaos.
We can use a computer to produce a picture of which magnet the pendulum will be attracted to. The magnets are located at the centers of the corresponding three large vase-shaped blocks of color. If you start the pendulum over a black region, it will eventually settle over the black magnet. Similarly, if you start the pendulum over a gray or white region, it will end up at the gray or white magnet. You can see regions of this picture in which moving the pendulum’s starting point a little won’t affect the outcome dramatically. For example, if you start near the black magnet, the pendulum is likely to end its journey at the black magnet. But there are other regions where the colors change rapidly over small distances.
This is an example of a shape that nature likes very much—the fractal. Fractals are the geometry of chaos, and if you zoomed in on some of the regions of this picture, you would see the same level of complexity, as we found on page 85. It is this complexity that makes the motion of the pendulum so hard to predict, even though the equations describing it are quite simple.
Figure 5.7 This computer-generated image illustrates the behavior of the pendulum moving over the magnets.
What if it’s not just the outcome of a swinging pendulum, but the future of the solar system that’s at stake? Perhaps a small perturbation by a rogue asteroid will cause a change that’s small but enough to send the solar system spiraling apart. This seems to have happened in the nearby solar system of the star Upsilon Andromedae, where astronomers believe that the strange behavior of the existing planets is evidence of a catastrophe in which one of the original planets orbiting the star was ejected after something disturbed the previously stable orbits. Could the same thing happen to our planet?
Just to reassure themselves, scientists have recently used supercomputers to try to answer the question that ultimately defeated Poincaré: is the earth really in danger of flying off into space? They ran the actual orbits of the planets backward and forward in time. Fortunately, the calculations showed that, with a 99 percent probability, the planets will continue to run smoothly in their orbits for another five billion years (by which time the sun will have evolved into a red giant star and swallowed up the inner solar system). But that still leaves a 1 percent chance of an outcome that’s slightly more interesting—at least mathematically.
It turns out that the rocky inner planets (Mercury, Venus, Earth, and Mars) have less stable orbits than the gas giants (Jupiter, Saturn, Uranus, and Neptune). Left to their own devices, these big planets would have a remarkably stable future. It’s tiny Mercury that has the potential to cause the solar system to go into catastrophic meltdown.
Computer simulations reveal that a strange resonance between Mercury a
nd Jupiter might evolve, which could cause Mercury’s orbit to start crossing the orbit of its nearest neighbor, Venus. That would set the stage for a potential almighty collision between Venus and Mercury, which would probably rip the solar system apart. But will this really happen? We don’t know. Chaos makes predicting the future very hard.
HOW CAN A BUTTERFLY KILL THOUSANDS OF PEOPLE?
It’s not just the solar system that’s chaotic. Many natural phenomena show chaotic traits: the behavior of the stock market, the buildup of a freak wave at sea, the beating of the heart. But the chaotic system that impacts everyone’s lives the most is the weather. “Will the earth still be orbiting the sun in a billion years’ time?” is not such an immediate concern. We want to know whether it will be warm and sunny next week, and ultimately whether the climate in 20 years’ time is going be dramatically different from its current state.
Weather forecasting has always been something of a dark art, even though some of the folklore to do with the weather has now been proved true. “Red sky at night, shepherd’s delight” works because the sun’s rays have been reddened by travelling through a large region of clear skies to the west of the shepherd. As weather systems in Europe generally arrive from the west, this signals good weather on the way.
Today’s meteorologists have a multitude of data to work with, ranging from measurements by weather stations at sea to images and information from satellites. And they have immensely precise equations to describe how the clashing air masses in the atmosphere interact to create clouds, wind, and rainfall. If we have the mathematical equations that control the weather, then can’t we just run the equations with today’s weather data through a computer and see what it will be like next week?
The Number Mysteries: A Mathematical Odyssey through Everyday Life Page 21
