The Physics of Superheroes: Spectacular Second Edition

by Kakalios, James

  If Ant Man is such a lightweight that he could be propelled across several city blocks by a coiled spring and not harm the ants that stopped his motion, then how is he able to disable such foes as the Protector or the Hijacker, or meet “The Challenge of Comrade X”? In particular, how is Ant-Man able to punch his way out of a vacuum-cleaner bag (shown in fig. 15), as thrillingly rendered in Tales to Astonish # 37, or swing a crook overhead using a nylon lariat in the very next issue? As explained in Tales to Astonish # 38, Henry Pym retained “all the strength of a normal human,” even when ant-size. Not to nitpick, but the average normal-size human, not to mention the average biochemist, would be hard-pressed to swing a full-grown man overhead, even using a “practically unbreakable” nylon lasso. But leaving that issue aside, what does it mean to say that Ant-Man has the strength of a normal-size person, such that he can break a vacuum bag, but only the mass of an ant, whereby he is easily sucked up by the vacuum cleaner in the first place? Perhaps the more basic question is: Why do you have the strength that you do, in that you can easily lift a twenty-pound object, but struggle with one weighing two hundred pounds and cannot possibly lift two thousand pounds? Our strength comes from our muscles and skeletal structure that comprise a series of interconnected levers. It turns out that these levers are not that well suited to lifting things.

  Let us stipulate that by “strength” (of its many definitions) we mean the ability to lift an object. Mankind’s ingenuity has led to the development of a wide range of machines to perform tasks such as heavy lifting. One of our earliest inventions employed to lift objects is the simple mechanical device of a lever. Many of us first encountered a lever as children in the form of a playground’s teeter-totter or seesaw, consisting of a horizontal board supported by a fulcrum point placed beneath the midpoint of the board. When seated at one end of the seesaw, you are able to lift a play-mate high in the air, through the mechanical advantage of the lever. With the fulcrum placed at the exact middle of the board, you can lift only a mass roughly equal to your own. If, however, the fulcrum point is placed much closer to one end, then a small child can lift a full-grown adult, provided the adult sits at the end of the seesaw nearer to the fulcrum point. This is because seesaws, and levers in general, do not balance forces but rather “torques.”

  Fig. 15. A scene from “Trapped by the Protector” from Tales to Astonish # 37, in which it is demonstrated that Ant-Man is both as light as an ant (and hence easily captured by a vacuum cleaner) and as strong as a normal-size human (and consequently able to punch his way out of the vacuum bag).

  If a force is defined as the ability to push or pull an object in a straight line, then a torque is a measure of the ability to rotate an object. A torque is mathematically defined as the product of the applied force and the distance between the force and the point where the object is to be rotated. While both “torque” and “work” are defined mathematically as the product of force and distance, in the case of work, the distance is the displacement of the object—that is, the distance over which the force pushes or pulls the object (more on work, in Chapter 12). The force must be acting in the same direction as this distance in order to change the object’s energy. In contrast, for a torque that causes a twist, the force should be at a right angle to the separation between the applied force and the point about which the object is to be rotated. This distance is referred to as the “moment arm” of the torque. For a given applied force, the larger the force’s distance from the point where the object is to be rotated, the greater the torque.

  This is why doorknobs are placed at the end of the door as far away from the hinges as possible. Try closing a door by pushing it at the end that’s immediately adjacent to the hinges, and then apply the same force to the other end, where the doorknob is located. The same force is used, but increasing the moment arm by increasing the distance between the push and the hinges magnifies the torque, and makes closing the door much easier. A wrench is another simple machine that amplifies a force applied at one end to produce a rotation at the other. When trying to loosen a particularly stubborn nut, one sometimes makes use of a “cheater,” basically an extension arm for the wrench by which the moment arm, and hence the applied torque, can be increased when the available force that can be applied is already at a maximum. Returning to the example of the seesaw, a small child is able to lift a full-grown adult only when the fulcrum of the seesaw is placed near the adult’s end (in a playground seesaw, the adult usually sits closer to the fixed fulcrum in the center). In this case, the moment arm for the child is increased, and the torque she applies is large enough to lift the adult up into the air, which the child could not accomplish without the mechanical advantage provided by the lever.

  Levers also play a role in determining the strength of Ant-Man’s tiny punch. Our arms are able to lift and throw by making use of the principle of levers. An object, let’s say a rock, is placed on one end of the lever, which we’ll call a “hand.” A force is exerted by the compression of the bicep muscle, causing the other end of the lever (the forearm) to move down, which in turn raises the far end of the lever—that is, the hand holding the rock. The biceps pulls the hand upward—when we need to lower the rock, the triceps contracts and in so doing pushes the hand back downward. Muscles can only contract and pull; they cannot push. Consequently, an ingenious series of levers, consisting of muscles attached to various points of our skeletal structure, have evolved in order to enable a wide range of movement. The fulcrum of the lever that is your forearm is located at the elbow. It may seem odd to have both forces applied on the same side of the fulcrum, but this type of lever is essentially the same as a fishing rod, where the force applied to one end—very near the fulcrum located near the reel—causes a rotation and consequent lifting of a fish at the other end of the rod. Your bicep applies a pulling force approximately two inches in front of your elbow, and most people’s forearms are fourteen inches long. The ratio of moment arms is thus 1:7, which means that the force applied by your biceps is reduced by a factor of seven at the location of your hand. That’s right, reduced—in order to lift a rock weighing 20 pounds, your biceps has to provide a lifting force of 140 pounds.

  A reasonable response to this news would be: What’s the point in that? Why have a lever built into your arm that increases the force needed to lift an object? There wouldn’t seem to be any point at all if the primary function of our arms were to lift rocks. Because the bicep is attached much closer to the fulcrum point (the elbow) than your hand, the biceps contracts two inches and the hand rises fourteen inches, due to the same ratio of 1:7 in moment arms. This ratio also holds when we want to get rid of this rock we are holding. In this case a muscle contraction of less than two inches produces a displacement of the hand of roughly twelve inches. This requires only 0.1 seconds to occur, and the hand holding the rock can get rid of it with a velocity of 12 inches in 0.1 second, or 10 feet per second (that is, 7 mph). This is a low estimate, and the average person can provide a much larger release velocity using other levers connecting her upper arm to her shoulder. A very, very small subset of the general population can throw baseball-size objects at speeds of up to 100 mph. And that’s the point of the inverse lever in our arms—it’s not intended to lift up large rocks; it’s there to enable us to throw smaller rocks at high velocities. Those of our ancestors who were better rock- or spear-throwers were, on average, better hunters. Being a better hunter increased one’s chance of securing dinner, and that in turn increased the odds of getting a date. In this way, certain hunters were able to pass these “good throwing-arm” genes down to their progeny.

  Meanwhile ... I haven’t forgotten about Ant-Man trapped in a vacuum-cleaner bag. For the tiny crime-fighter, all length scales are obviously reduced, but the ratio of moment arms of 1:7 in his arms still holds for Henry Pym, regardless of whether he is ant-size or normal height. Punching involves the same muscles and similar motions as throwing, only instead of a rock, one is throwing a fist. The force provide
d by your muscles does not depend on their length, but on their cross-sectional area (that is, the area one would measure in a magnetic resonance imaging slice though the thickness of the bicep, not the surface area around the outside of the arm). If Ant-Man is 0.01 times his normal height, then the force his muscles can provide is reduced by a factor of (0.0 1)2 = 0.0001. If Pym can punch with a force of two hundred pounds when full size, his scaled-down punch delivers a wallop of two hundredths of a pound. At his miniature size, his fist is much smaller and has a cross-sectional area of 0.0005 square inches (assuming his hand is just a millimeter wide). The pressure that his punch provides is defined as the “force per unit area,” which is 0.02 pounds divided by 0.0005 square inches—or 40 pounds per square inch.26 This is to be compared with his normal- height punching force of 200 pounds divided by his normal-size fist’s cross-sectional area of 5 square inches, for a pressure of 40 pounds per square inch. That is, Henry Pym’s punches exert the same pressure when he is ant-size as they do when he’s at his normal height. If he can punch through the vacuum bag while normal sized, then he can do so at his reduced height. It appears that Ant Man can indeed punch his way out of paper bag. In this way, he serves as a role model and inspiration to all comic-book fans.

  WHY BEING BITTEN BY A RADIOACTIVE SPIDER ISN’T ALL IT’S CRACKED UP TO BE

  While we’re on the subject of one’s strength while the size of an insect, I would like to take a moment to dispel a myth concerning Spider-Man. As we have just argued, if Henry Pym shrinks at a constant density, then while the force of his punch is not as great as when he is at normal height, the pressure his fist is able to supply to an unsuspecting vacuum-cleaner bag is unchanged. A common misconception is that this scaling works in both directions, so that if one were to be bitten by a radioactive spider, just to take a random example, then one would gain the proportionate leaping ability of a spider. That is, if a spider or flea can jump one meter high—which is roughly 500 times higher than its body height—then a human with a comparable leaping ability would be able to leap a distance roughly 500 times his body height. For someone six feet tall, this implies a jumping range of 3,000 feet! If this were indeed the case, then Spider-Man would have the Golden Age (preflight, pre-yellow-sun-derived superpowers) Superman beat by—and here the expression could be taken nearly literally—a country mile. However, this is nowhere near the case. If Peter Parker did indeed gain the leaping ability of a spider, then he would be able to jump the same distance as a spider—namely, one meter. For the sake of exciting and engaging comic-book stories, it is a good thing that Stan Lee and Steve Ditko did not understand this scaling problem. Let’s see where they went wrong.

  What determines how high you can leap? Two things only: your mass and the force your leg muscles can supply to the ground. These two factors determine how much acceleration you can achieve as you lift off the ground. Once you are no longer in contact with the pavement, the only force acting upon you is gravity, which slows you down as you ascend. So there are two accelerations we have to concern ourselves with: the initial liftoff boost that gets you airborne, and the ever-present deceleration of gravity that eventually halts your rise. Once you are moving with some large velocity v, the height h you will climb is given by the familiar formula from before v2 = 2gh, where once again g represents the deceleration due to gravity.

  There is a surprising aspect of this equation that we have not yet remarked upon—namely, that nowhere does the final height that the leaper reaches depend on the mass of the person jumping! Big or small, if you start off with a velocity v and the only thing pulling you back to Earth is gravity, then your eventual height depends only on the deceleration due to gravity g and your initial velocity v. Of course, there is another acceleration that enters into the leap—that provided by your leg muscles at the start of the jump. And this acceleration does depend on the mass of the leaper. Using Newton’s second law of motion, that force equals mass times acceleration, or F = ma, it is clear that for a given force F, the larger the person (that is, the bigger his mass m), the smaller will be his acceleration a, and the less of an initial liftoff velocity he will achieve. A smaller starting velocity means a lower height h you will be able to jump.

  It’s not that spiders are such great leapers that they can jump many times their body length. Rather, it’s that small insects have tiny muscles (providing a small force), but they only have to lift an equally tiny mass to leap one meter, which just turns out to be many times larger than their size. Humans have much bigger muscles than spiders and can achieve much greater forces, but they have to lift much greater masses, so the net effect is that the height they can jump is also about one meter. Of course, some humans such as Olympic high jumpers can leap much higher than this, while most of us slugs can jump barely more than a third of a meter (that is, one foot). In fact, for a flea to leap two hundred times its body length requires a great deal of cheating on nature’s part: In addition to being particularly streamlined to minimize air drag, the flea pushes off with its two longest legs to maximize the lever arm. These are its hind legs, so in fact fleas always jump backward when they alight.

  It is a natural mistake when scaling up the abilities of the insect and animal kingdoms to human dimensions to assume that it is the proportions that are important, rather than the absolute magnitudes. In the nineteenth century, many distinguished entomologists made the same error. As succinctly put in a footnote in the classic On Growth and Form by D’Arcy Thompson: “It is an easy consequence of anthropomorphism, and hence a common characteristic of fairy-tales, to neglect the dynamical and dwell on the geometrical aspect of similarity.” But such misconceptions make for much more interesting fairy tales, and comic-book stories.

  9

  THE HUMAN TOP GOES OUT FOR A SPIN—ANGULAR MOMENTUM

  WHEN ROSCOE DILLON was entering adolescence, he developed an obsession that would consume his life as an adult. What makes Roscoe unique is that unlike most young teenagers, his fixation involved spinning tops. Whether it was a simple children’s toy or a sophisticated gyroscope, Roscoe was fascinated by tops. But he gave up playing with them as an adult, when he drifted into an unsuccessful life of crime. During his second stint in prison, he concluded that he needed a gimmick in order to be a successful thief, and saw that his youthful love of tops could be just the novel angle that his criminal career lacked. Upon release from the penitentiary, he studied all aspects of top rotation, and constructed special tops that emitted gas bombs, bolos, or entangling streamers, with which he would embark on what initially appeared to be a successful crime spree. Naturally he committed these thefts wearing a bright green unitard accented with narrow horizontal yellow rings running along the length of his body—the better to strike fear into the hearts of men. He also trained himself to spin rapidly about an axis passing through the length of his body, whereby he discovered that “the spinning action increases my brain power!”27

  Of course, having set up his base of operations in Central City, he soon attracted the attention of the Crimson Comet, as relayed in “Beware the Atomic Grenade” (always sound advice) in Flash # 122 from August 1961. Initially the Top was able to escape from the Scarlet Speedster, but he eventually overplayed his hand. The Rotating Rogue managed to trap the Flash inside a giant “atomic grenade” that spun about its central axis like a top. Dillon threatened that unless all of the world’s governments made him, the Top, the supreme ruler of the world (which is quite a jump from the beginning of this comic-book story, where we meet Dillon stealing the payroll from Wimbel’s Department Store (note to lawyers—n ot to be confused with the major 1960s retailer Gimbels department store), his atomic grenade would explode and destroy half of the planet. In case the world’s governments did not accede to his demand, Dillon had a backup plan. The Top planned on relocating to the other side of the globe, in North Africa, where he would be “safe from harm!” The devastation that would visit his “safe” side of the Earth if the other half were obliterated by atomic d
evastation seems to not have occurred to this “mentally enhanced” master criminal. Not that you should be worried, Fearless Reader, as the Flash eventually managed to run at such a great speed around the atomic grenade that he built up a mass of compressed air that launched the bomb at escape velocity,28 never to return to Earth. He then searched the other hemisphere until he located the Top, and delivered him in to the Central City police. In fact, Flash stories in the 1960’s often ended with the perpetrators of attempted global terrorism being handled as a local, municipal matter. It should be noted that all of the Top’s battles with the Flash ended this same way—with the Top behind bars. In appears that Dillon’s fixation with tops was not exactly the key missing ingredient that would propel him to a successful crime career.