In fact, this problem is found throughout the universe. There are regions of space that contain gases that emit X-rays that are very hot—one million degrees hot. These gases collect around galaxies, yet once again there is not enough mass in the observable stars to provide sufficient gravity to hold these hot gases in place. Astronomers have concluded that there is not enough mass in stars, interstellar dust, or planets to account for the gravitational stability of these hot gases. There must be another, unseen source of gravity that we cannot yet detect.
There are only two options—either we don’t really understand gravity (which is possible, although the current theory of gravity, expressed in Einstein’s General Theory of Relativity, has been experimentally verified enough times for us to have some measure of confidence in it), or there is a lot of mass in the universe that we just can’t see. Scientists have concluded that the second option is the more palatable, and there is general agreement that nearly all of the universe’s mass is actually contained in what has been termed Dark Matter. In fact, as will be discussed in Section 2, there is in fact a precedent, dating back to the 1930s, for inventing a “miracle mass” to avoid having to throw out all previously accepted and verified physics principles. As of this writing, scientists do not know anything, really, about the composition of this ubiquitous material. Everything we see in a clear night sky, the billions of galaxies, each cluster containing billions of stars—is nothing more than a few percent of all of the mass and energy in the universe. The Top and Whirlwind aspired to be master criminals, but they never dreamed that the source of their superpower—angular momentum—could lead us to conclude that 96 percent of the universe is missing. The ultimate crime, indeed.
10
IS ANT MAN DEAF, DUMB, AND BLIND?—SIMPLE HARMONIC MOTION
THERE ARE SEVERAL SUPERHEROES whose special power is the ability to shrink. In addition to Ant Man and the Wasp in the Marvel universe, the Atom, Elasti-Girl of the Doom Patrol, and Shrinking Violet of the Legion of Super-Heroes in DC Comics all share the ability to miniaturize themselves. The “explanation” for their shrinking powers varies, but the laws of physics dictate one certainty about these tiny superheroes: They will find it very difficult to communicate. I’m not referring to Ant Man’s and the Wasp’s relationship problems that led to their divorce, but rather the physical limitations involving conversations with anyone in the non-shrunken world.
If you’re only a few millimeters tall, no one can hear you, nor will your own hearing be too keen, so you will have to depend on nonverbal means to communicate with normal-size people. As Ant Man shrinks, his voice becomes higher pitched, until when he has reached the size of an ant, his normal speaking voice will be near the upper range of normal-size-human hearing. At the same time, his hearing threshold also shifts to the higher end of the register, such that he will miss most of what people are saying to him. To make matters worse, everything our tiny hero sees will be an out-of-focus blur. Let’s see why being an “inch high private eye,” while not cutting you off completely from the outside world, will nevertheless lead to a host of problems.
First we address the basic question: What determines the range at which we are able to speak and hear? To answer this, let’s look at a pendulum. A pendulum is just a mass connected to a thin string (we’ll ignore the weight of the string) with the other end connected to a frictionless pivot point on the ceiling. The mass is usually assumed to be some dense sphere, such as a billiard ball or a bowling ball, but it could be a rock or Spider-Man. This mass is lifted to some height, such that the string makes a small angle with the vertical direction and, once released, the forces acting on it are (1) gravity, always pulling straight down in the vertical direction, and (2) the upward tension in the string. The tension’s direction makes an angle relative to the vertical and is continually changing as the mass swings back and forth. Part of the bob’s weight can be viewed as directed along the line of the string, while the part of the weight that is not compensated by string’s tension is the force that changes the mass’s speed and is responsible for the acceleration of the mass as it moves to and fro.
The time that a pendulum takes to move back and forth—from some high initial point, through a complete arc, and back to its original position—is called the “period,” and its motion is called “periodic.” Whether it’s Spider-Man swinging back and forth on his webbing, or a billiard ball attached to fishing line, the time to complete a full oscillation depends on only two factors: the acceleration due to gravity (what we have called g) and the length of the string. Surprisingly, one thing that the period does not depend on (at least for small angles of oscillation) is the initial height at which the mass starts its motion.
Galileo was perhaps not the first person to notice that the period of a pendulum is an intrinsic property and independent of how high or low the swinging mass’s starting point is, but he is properly credited for determining what controls the oscillation rate. As surprising and counterintuitive as it seems, the time necessary for a playground swing to go back and forth depends neither on how heavy or light the person sitting on the swing is, nor on how far back he starts his motion, but only on the length of chain between the seat of the swing and the top pivot point of the swing set. We are assuming that the swing is not being pushed by a stationary helper nor is the person in the swing pumping his legs as he moves back and forth. It is certainly true that the higher the starting position of the swing, the faster you will be moving at the bottom point of your swing, since the component of the tension in the string that deviates from the vertical (responsible for accelerating along the swing) is greater the larger the starting angle. Shouldn’t this mean that it should take less time to complete an oscillation? No, because while you may be moving faster owing to your higher initial position, you have farther to travel until you reach that bottom point of the arc. The combination of the greater speed but longer distance to be traversed balances out, such that the time needed to complete the arc remains the same, regardless of the starting point. This is why a pendulum or any other device undergoing simple harmonic motion makes a great timekeeper. Two identical clocks, using either a swinging bob like that in a grandfather clock, or a coiling and uncoiling spring like that found in old-fashioned pocket watches or metronomes, will keep identical time, independent of the starting push that begins the oscillation. A metronome is an upside-down pendulum, and its frequency is independent of how it is set in motion, but it can be altered by changing the location of the mass on the swinging arm (which has the effect of changing the length of the pendulum).
If the period of a pendulum does not depend on the starting point of the swing, why does it depend on gravity and the length of the string? It’s not hard to see that the weaker the acceleration due to gravity, the less force there will be pulling on the mass and the slower will be its motion. On the moon a pendulum will take longer to complete a cycle than it does on Earth, and in deep space, where we may take the acceleration due to gravity g to be zero, the bob will never move at all. Why does the length of the string enter into the pendulum’s period? Because of torque. There is a torque rotating the bob along its trajectory—the force is the component of the bob’s weight along the arc and the moment arm is the length of the pendulum’s string. The mass shows up on both sides of the rotational form of F= ma, and cancels out. The two remaining factors, the acceleration due to gravity and the moment arm l determine the time necessary to complete a cycle.
The frequency of the pendulum—the number of back-and-forth oscillations it completes in one second—is just the inverse of the period, defined as the time needed to finish one cycle. If the period is one tenth (0.1) of a second, then it will have a frequency of ten cycles per second. The shorter the period, the higher the frequency. The square of the period, in turn, is proportional to the ratio of the length of string in the pendulum l and the acceleration due to gravity g. That is, the (period)2 = (2π)2 × (l/g). To explain why it is the square of the period, and not just
the period, that depends on the ratio of l over g, and why the factor of (2π)2 enters, would require us abandoning our “algebra-only” pledge. For our purposes, the important point is that one has to increase the length of the pendulum’s string by a factor of four to double the period. Conversely, shrinking the length of the string (say, by using Pym particles) decreases the period, and the smaller the period, the higher the frequency.
Speaking involves the generation of sound waves, which are created by forcing air passed vibrating vocal cords. A human vocal cord is not a mass swinging back and forth on a string, but the beauty of the pendulum as a description of simple harmonic motion is that it captures the important physics of any oscillating system.31 When Henry Pym shrinks down to ant size, he reduces his dimensions by roughly three hundred times. The fundamental frequency of the oscillator is, correspondingly, seventeen times bigger (that is, by the square root of three hundred). Normal human speech occurs at a pitch of roughly two hundred cycles per second, but for an ant-size person the frequency is shifted up by this factor of seventeen, to 3,400 cycles per second. Our hearing range extends from twenty cycles per second at the low end up to twenty thousand cycles per second, so we should still be able to hear Ant Man, but he will have a high-pitched voice, as his chest cavity similarly shrinks. When they hear a quarter-of-an-inch-tall superhero order them to surrender in such a squeaky voice, it is surprising that Ant Man’s foes do not succumb to fits of laughter rather than his tiny right hook.
Not only will his voice change as he shrinks, but Ant Man’s hearing will also be affected by his reduced height. The resonant frequency of a drum also rises as its diameter shrinks. A large bass drum has a deep, low tone, while a smaller snare drum emits a higher pitch when struck. When Dr. Pym’s eardrums shrink upon exposure to Pym particles, the frequencies that he is able to detect shift accordingly. (The physics underlying the range of human hearing is actually pretty complicated, but for our purposes we’ll assume that it is determined by the eardrum.) The lowest frequency he can hear when at his normal six-foot size is roughly twenty cycles per second, which upon shrinking becomes seventeen times greater at nearly 340 cycles per second. A normal person’s speech, at a pitch of two hundred cycles per second, will therefore be below the range of detection for our tiny titan. For this reason, Ant Man and his miniaturized colleagues will need to be astute students of body language as they interact with the normal world.
In addition to the change in the threshold frequency of his eardrums, Henry Pym’s hearing sensitivity will also be affected as he shrinks to the size of an ant. As the vocal cords vibrate back and forth, they cause alternating compressions and expansions on the air rushing past them, forced through the throat by contractions of the diaphragm. This variation in density is slight—only one part in ten thousand on average distinguishes the adjacent regions of compressed and rarefied air. The larger the density variation, the greater the volume or loudness of the sound wave. You only have control over the initial density variation as the sound leaves your mouth. The compressed region of air expands, compressing the region in front of it, which in turn expands and squeezes the next region of air. What you hear is the instruction set for the sound wave, generated from your vocal cords and transmitted to your ears. The air from your mouth does not physically travel from you to the listener. If I tell you that I had garlic for lunch, you hear this information before you come close enough to receive independent confirmation of this fact. As the information is spreading out in all directions, the farther the hearer is away from the speaker the more the variation in air density—the sound wave—will be attenuated. Beyond a certain point the information will not be detectable at all.
Alternatively, if one is too close to the source, the eardrum is unable to linearly respond to the density variations, and the ability to distinguish differing sounds is degraded. This can be handy, as the DC Comics miniaturized hero the Atom discovers in “The Case of the Innocent Thief” in Atom # 4.* In this story, a crook named Elkins discovers a hypnotic ray that compels people to obey his oral commands. While the Atom is only a few inches tall, the crook exposes him to this ray and yells a command forbidding the Atom to try to capture him. Yet almost immediately the Atom knocks out Elkins, using a pink rubber eraser as a trampoline to come within punching distance. The Atom is able to resist the mesmerizing command because, as he explains at the end of the story, “in his excitement, Elkins shouted his orders at me—words which sounded like thunder to me! And since I couldn’t understand a word he said to me, I didn’t have to obey him!” (Interestingly enough, my kids put forth this same argument nearly daily, though they have not yet mastered miniaturization technology, nor am I shouting.)
If the hearing and speaking issues weren’t bad enough, Ant-Man’s vision should also be blurred. The average spacing between adjacent peaks or valleys (that is, the wavelength) in the alternating electric and magnetic fields that comprise a light wave determines the color of the light. Let us say that on average, white light—which consists of light of all wavelengths from red (650 nanometers) to violet (400 nanometers) added together in equal magnitudes—has a wavelength of 500 nanometers (one nanometer is one billionth of a meter). In order for light to be detected, it must hit the rods and cones in the back of your eye, and in order to get to these photoreceptors, it must first pass through your pupil. This opening in the front of your eye, depending on the brightness in the room in which you are reading this, is roughly five millimeters in diameter. One millimeter is equal to a million nanometers, so the opening in your pupil is roughly ten thousand times larger than the wavelength of visible white light. From the point of view of the light waves, the pupil is32 a very large tunnel through which they can easily pass. When Ant Man shrinks to the size of an insect, however, the opening in his pupil will be three hundred times smaller than it was at his normal height. The orifice in his eye is now roughly a factor of thirty times larger than the wavelength of visible light, which is still 500 nanometers. The light waves can still fit into this “tunnel,” but just barely.
To understand what the consequences are when the size of the opening is only a few times bigger than the wavelength of light, consider water waves on the surface of a large lake. A channel is formed between two docks that ride low on the surface of the water. When the separation of the docks is very large, say half a mile apart, compared with the spacing between peaks of the water waves, the waves pass through this region with no noticeable perturbation. Right near the dock, as the waves break, there is a change in the wave front, but in the middle between the docks the waves are hardly affected by the docks. This is the situation for Henry Pym at his normal height, when his pupil is ten thousand times larger than the wavelength of light. For the miniaturized Ant-Man, it is as if the two docks narrow to a bottleneck, so that the separation of the docks is only a few times more than the separation between adjacent wave peaks. The waves still move through the constriction, but as they scatter off the edges of each dock they set up a complicated interference pattern on the other side of the obstruction. This effect is termed “diffraction” and is most noticeable when the dimensions of the object scattering a wave are comparable to the wavelength. If you were hoping to gain information about the cause of the water ripples by examining the wave fronts, you would have obtained a clean, sharp image when the docks were thousands of feet apart and a distorted and confusing picture when the docks were only separated by a few feet.
The effect for Ant-Man is that the image he observes through his shrunken pupil will be blurry and out of focus. This is why an insect’s eye, and in particular its lens, is radically different from the lens in a human’s or larger animal’s eye. Insects use compound lenses that adjust for the diffraction effects. Even so, it would be hard for a fly to read a newspaper, even if he did care about current events. An insect’s eye is very good at detecting changes in light sources (such as the moving shadow created by the rolled-up newspaper of doom), but poor at discerning the contrast between sh
arp edges. Consequently they rely on other senses, such as smell and touch (hair filaments detect subtle variations in air currents) to navigate their way through the wide world. Unfortunately for Ant Man, the one sense that is least affected by miniaturization, smell, is the one that is the least sensitive in humans.
11
LIKE A FLASH OF LIGHTNING—SPECIAL RELATIVITY
IN AN EARLIER CHAPTER I MENTIONED the sonic boom that the Flash creates whenever he runs faster than the speed of sound. Why is there a “boom” when an object moves at or faster than the speed of sound? And how can this help us understand Einstein’s Special Theory of Relativity?
First, let’s consider the boom, and then get to Einstein. Imagine yourself standing out in the countryside, and the Flash is running toward you at the speed of sound—that is, at one fifth of a mile per second. If he starts ten miles away from you, he’ll reach you in fifty seconds. When he is ten miles away, he says, “Flash” and when he is only five miles in front of you, he says, “Rules.” What do you hear? If the Flash was running slower than the speed of sound, then the “Flash” he spoke would reach the five mile mark before he did, and then he would utter “Rules” while the “Flash” was about to reach your ears. You would clearly hear “Flash Rules,” followed a short while later by the sound of the Scarlet Speedster running past you.
The Physics of Superheroes: Spectacular Second Edition Page 14