The Physics of Superheroes: Spectacular Second Edition (18 page)

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 v
²
= 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.

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 devastation 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.

DC Comics was not the exclusive home of rotating supervillains—Marvel’s Giant-Man was bedeviled by his own whirling dervish: Dave Cannon, also known as the Human Top. Cannon was a mutant whose superpower involved the ability to spin at high speeds. In his first appearance in 1963’s
Tales to Astonish # 50,
the Human Top battled Giant-M an and the Wasp when they attempted to stop his theft of the payroll of Danly’s Department Store (the elucidation of the basis for the attraction between spinning supervillains and department-store payrolls remains a great open question in science). One difference between DC Comics’ Top and Marvel’s Human Top, aside from the fact that Marvel’s villain’s code name proudly proclaims his species (and while the Top was originally just a guy who liked to spin, the Human Top was in fact a mutant), is that Cannon never claimed that his spinning invoked any improvement in his mental faculties. In fact, he was considered so irresponsible that he was dropped from the Masters of Evil by Egghead. When other villains do not consider you up to the same high standards exhibited by the Beetle and the Shocker, you really should take a long look at your life. Cannon modified his costume, adopting a helmet that looked like the top half of a bright green missile with two broken hockey sticks protruding from it, going shirtless, and changing his moniker to Whirlwind. Shockingly, the addition of a green pointy helmet did not earn Whirlwind any new respect.

The difficulties that the Top and Whirlwind had breaking into the upper echelons of villainy is surprising, for their superpower involves one of the most important concepts in physics—Angular Momentum. We have discussed “linear momentum” in Chapter 3 as part of our description of the physics underlying the death of Gwen Stacy. There we did not need the modifier “linear,” because only one type of momentum was discussed—the straight-line kind. Now we’ll consider a more general situation where an object rotates or orbits about an axis. Linear momentum is simple, as there is only one way that an object can move in a straight line, but there are in principle an infinite number of axes that a mass can revolve about. The axis of rotation could pass through the spinning object, as in the case of the Top and Whirlwind, who twirl about an imaginary line passing through their body from their feet to the top of their head. But the axis of rotation need not pass through the volume of the object, such as when the mighty Thor twirls his mystical hammer Mjolnir. In this case, the axis of rotation is a line at a right angle to the plane formed by the spinning hammer, along the length of Thor’s outstretched arm. Alternatively, the axis of rotation could be a great distance from the object in question, as in the Earth orbiting the sun, where the trajectory of our planet defines a disc (technically an ellipse), with the axis of rotation again being perpendicular to this plane and passing through the sun.

The Principle of Conservation of Linear Momentum states that a mass moving in a straight line will continue to do so if no outside forces act upon it—and that when such forces are present the change in momentum can be related to the product of the force and the time it acts, which we called the Impulse in Chapter 3. For Gwen Stacy, knocked from the top of the George Washington Bridge by the Green Goblin, her momentum steadily increased as she fell due to the external force acting upon her—namely, gravity. In order to change this momentum when Spider-Man stopped her descent, he needed to apply another force, through his webbing.

Physics has identified a corresponding Principle of Conservation of Angular Momentum that has important similarities to the Conservation of Linear Momentum. Linear momentum is mathematically defined as the product of an object’s mass and velocity. The inertial mass reflects how difficult it is to change the motion of the object—it is easier to deflect a mosquito than a Mack truck. For a given mass, the larger its velocity, the greater its momentum and the more force is needed to alter its motion.

Similarly, an object’s angular momentum is mathematically defined as the product of its “moment of inertia”—which reflects how difficult it is to rotate the object about a particular axis—and its rotation speed. An object rotating about an axis passing through itself or orbiting about some external axis will continue to do so, unless some outside force acts to change its rotation.

It is easier for an ice skater to twirl about an axis passing along the length of her body, the way the Top or Whirlwind does, if her arms are at her sides. Extending her arms away from her body places more mass further from the axis of rotation, and increases her moment of inertia. The more weight distributed away from the axis of rotation, the harder it is to spin. In this case we say that the moment of inertia is larger. For a given object, the faster the rotation speed, the greater the angular momentum. The harder it is to rotate an object at a given rotation speed, the harder it is to stop its spinning, and the larger the angular momentum.

An object moving in a straight line has a constant momentum, if no outside force acts on it. Thus, if it were to lose some of its weight, its speed would have to increase in order to keep the product of mass and velocity constant (that is, the linear momentum unchanged). This is the principle by which rockets and jets fly (hey—it
is
rocket science after all!). Similarly, if a spinning object changes its distribution of mass, then it alters how easy or difficult it is to rotate about an axis, that is, its moment of inertia. If there is no outside torque, the angular momentum cannot change, so an increase in the moment of inertia leads to a decrease in the rotation speed. When a figure skater wants to twirl at a faster rate, he or she pulls their arms in toward their body. Having more mass closer to the axis of rotation makes it easier to spin, and the decrease in the moment of inertia leads to a faster spinning rate. Whirlwind’s motivations in designing a helmet with partial hockey sticks protruding from the sides, moving mass away from the rotation axis and increasing his moment of inertia and thus making it harder for him to spin further demonstrates that he is not the sharpest knife in the supervillain kitchen drawer.