A famous urban legend states that a penny dropped from the top of the Empire State Building will punch a hole in the sidewalk below. Given the height of the building and the hardness of the penny, that seems like a reasonable possibility. Whether it's true or not is a matter that can be determined scientifically. Before we do that, though, let's get some background.
Falling rocks can be dangerous and, the farther they fall, the more dangerous they become. Falling raindrops, snowflakes, and leaves, however, are harmless no matter how far they fall. The distinction between those two possibilities has nothing to do with gravity, which causes all falling objects to accelerate downward at the same rate. The difference is entirely due to air resistance.
Air resistance—technically known as drag—is the downwind force an object experiences as air moves passed it. Whenever an object moves through the air, the two invariably push on one another and they exchange momentum. The object acts to drag the air along with it and the air acts to drag the object along with it, action and reaction. Those two aerodynamic forces affect the motions of the object and air, and are what distinguish falling snowflakes from falling rocks.
Two types of drag force affect falling objects: viscous drag and pressure drag. Viscous drag is the friction-like effect of having the air rub across the surface of the object. Though important to smoke and dust particles in the air, viscous drag is too weak to affect larger objects significantly.
In contrast, pressure drag is strongly affects most large objects moving through the air. It occurs when airflow traveling around the object breaks away from the object's surface before reaching the back of the object. That separated airflow leaves a turbulent wake behind the object—a pocket of air that the object is effectively dragging along with it. The wider this turbulent wake, the more air the object is dragging and the more severe the pressure drag force.
The airflow separation occurs as the airflow is attempting to travel from the sides of the object to the back of the object. At the sides, the pressure in the airflow is especially low due as it bends to arc around the sides. Bernoulli's equation is frequently invoked to help explain the low air pressure near the sides of the object. As this low-pressure air continues toward the back of the object, where the pressure is much greater, the airflow is moving into rising pressure and is pushed backward. It is decelerating.
Because of inertia, the airflow could be expected to reach the back of the object anyway. However, the air nearest the object's surface—boundary layer air—rubs on that surface and slows down. This boundary layer doesn't quite make it to the back of the object. Instead, it stops moving and consequently forms a wedge that shaves much of the airflow off of the back of the object. A turbulent wake forms and the object begins to drag that wake along with it. The airflow and object are then pushing on one another with the forces of pressure drag.
Those pressure drag forces depend on the amount of air in the wake and the speed at which the object is dragging the wake through the passing air. In general, the drag force on the object is proportional to the cross sectional area of its wake and the square of its speed through the air. The broader its wake and the faster it moves, the bigger the drag force it experiences.
We're ready to drop the penny. When we first release it at the top of the Empire State Building, it begins to accelerate downward at 9.8 meters-per-second2—the acceleration due to gravity—and starts to move downward. If no other force appeared, the penny would move according to the equations of motion for constant downward acceleration, taught in most introductory physics classes. It would continue to accelerate downward at 9.8 meters-per-second2, meaning that its downward velocity would increase steadily until the moment it hit sidewalk. At that point, it would be traveling downward at approximately 209 mph (336 km/h) and it would do some damage to the sidewalk.
That analysis, however, ignores pressure drag. Once the penny is moving downward through the air, it experiences an upward pressure drag force that affects its motion. Instead of accelerating downward in response to its weight alone, the penny now accelerates in response to the sum of two force: its downward weight and the upward drag force. The faster the penny descends through the air, the stronger the drag force becomes and the more that upward force cancels the penny's downward weight. At a certain downward velocity, the upward drag force on the penny exactly cancels the penny's weight and the penny no longer accelerates. Instead, it descends steadily at a constant velocity, its terminal velocity, no matter how much farther drops.
The penny's terminal velocity depends primarily on two things: its weight and the cross sectional area of its wake. A heavy object that leaves a narrow wake will have a large terminal velocity, while a light object that leaves a broad wake will have a small terminal velocity. Big rocks are in the first category; raindrops, snowflakes, and leaves are in the second. Where does a penny belong?
It turns out that a penny is more like a leaf than a rock. The penny tumbles as it falls and produces a broad turbulent wake. For its weight, it drags an awful lot of air behind it. As a result, it reaches terminal velocity at only about 25 mph (40 km/h). To prove that, I studied pennies fluttering about in a small vertical wind tunnel.
Whether the penny descends through stationary air or the penny hovers in rising air, the physics is the same. Of course, it's much more convenient in the laboratory to observe the hovering penny interacting with rising air. Using a fan and plastic pipe, I created a rising stream of air and inserted a penny into that airflow.
At low air speeds, the penny experiences too little upward drag force to cancel its weight. The penny therefore accelerated downward and dropped to the bottom of the wind tunnel. At high air speeds, the penny experienced such a strong upward drag force that it blew out of the wind tunnel. When the air speed was just right, the penny hovered in the wind tunnel. The air speed was then approximately 25 mph (40 km/h). That is the terminal velocity of a penny.
The penny tumbles in the rising air. It is aerodynamically unstable, meaning that it cannot maintain a fixed orientation in the passing airstream. Because the aerodynamic forces act mostly on the upstream side of the penny, they tend to twist that side of the penny downstream. Whichever side of the penny is upstream at one moment soon becomes the downstream side, and the penny tumbles. As a result of this tumbling, the penny disturbs a wide swath of air and leaves a broad turbulent wake. It experiences severe pressure drag and has a low terminal velocity.
The penny is an example of an aerodynamically blunt object—one in which the low-pressure air arcing around its sides runs into the rapidly increasing pressure behind it and separates catastrophically to form a vast wake. The opposite possibility is an aerodynamically streamlined object—one in which the increasing pressure beyond the object's sides is so gradual that the airflow never separates and no turbulent wake forms. A penny isn't streamlined, but a ballpoint pen could be.
Almost any ballpoint pen is less blunt than a penny and some pens are approximately streamlined. Moreover, pens weigh more than pennies and that fact alone favors a higher terminal velocity. With a larger downward force (weight) and a smaller upward force (drag), the pen accelerates to a much greater terminal velocity than the penny. If it is so streamlined that it leaves virtually no wake, like the aerofoil shapes typical of airplane components, it will have an extraordinarily large terminal velocity—perhaps several hundred miles per hour.
Some pens tumble, however, and that spoils their ability to slice through the air. To avoid tumbling, a pen must "weathervane"—it must experience most of its aerodynamic forces on its downstream side, behind its center of mass. Arrows and small rockets have fletching or fins to ensure that they travel point first through the air. A ballpoint pen can achieve that same point-first flight if its shape and center of mass are properly arranged.
Almost any ballpoint pen dropped into my wind tunnel plummeted to the bottom. I was unable to make the air rise fast enough to observe hovering behavior in those pens. Whether they would tend to tumble in the open air was difficult to determine because of the tunnel's narrowness. Nonetheless, it's clear that a heavy, streamlined, and properly weighted pen dropped from the Empire State Building would still be accelerating downward when it reached the sidewalk. Its speed would be close to 209 mph at that point and it would indeed damage the sidewalk.
As a final test of the penny's low terminal velocity, I built a radio-controlled penny dropper and floated it several hundred feet in the air with a helium-filled weather balloon. On command, the dropper released penny after penny and I tried to catch them as they fluttered to the ground. Alas, I never managed to catch one properly in my hands. It was a somewhat windy day and the ground at the local park was uneven, but that's hardly an excuse—I'm simply not good at catching things in my hands. Several of the pennies did bounce off my hands and one even bounced off my head. It was fun and I was more in danger of twisting my ankle than of getting pierced by a penny. The pennies descended so slowly that they didn't hurt at all. Tourist below the Empire State Building have nothing fear from falling pennies. Watch out, however, for some of the more streamlined objects that might make that descent.
When you carry the firewood up the hill, you transfer energy to it and increase its gravitational potential energy. When you then burn the wood, you seem to make this energy disappear. After all, there doesn't appear to be any difference between burning wood in the valley and burning wood on the top of the hill. The wood is gone either way.
But appearances can be deceiving. Since energy is a conserved quantity, the energy that you invest in the firewood can't disappear. It simply becomes difficult to find because it is dispersed in the burned gases that were once the wood.
To find that energy, imagine compressing the burned gases into a small container to make their weight more noticeable and reduces buoyant effects due to the atmosphere. You could then carry those burned gases, which include all of the firewood's atoms, back down the hill. As you descended, the container of burned gases would transfer its gravitational potential energy to you.
I've swept a number of details under the rug, such as the fact that many of the oxygen atoms in your container were originally part of the atmosphere rather than the log. But even when all those details are taken into account, the answer is the same: the firewood's gravitational energy doesn't disappear, it just gets more difficult to find.
The answer is gravity. Gravity smashes the planets into spheres. To understand this, imagine trying to build a huge mountain on the earth's surface. As you begin to heap up the material for your mountain, the weight of the material at the top begins to crush the material at the bottom. Eventually the weight and pressure become so great that the material at the bottom squeezes out and you can't build any taller. Every time you put new stuff on top, the stuff below simply sinks downward and spreads out. You can't build bumps bigger than a few dozen miles high on earth because there aren't any materials that can tolerate the pressure. In fact, the earth's liquid core won't support mountains much higher than the Himalayas—taller mountains would just sink into the liquid. So even if a planet starts out non-spherical, the weight of its bumps will smash them downward until the planet is essentially spherical.
The flattened poles are the result of rotation—as the planet spins, the need for centripetal (centrally directed) acceleration at its equator causes its equatorial surface to shift outward slightly, away from the planet's axis of rotation. The planet is therefore wider at its equator than it is at its poles.
Converting units is always a matter of multiplying by 1. But you must use very fancy versions of 1, such as 60 seconds/1 minute and 1 gallon/3.7854 liters. Since 60 seconds and 1 minute are the same amount of time, 60 seconds/1 minute is 1. Similarly, since 1 gallon (U.S. liquid) and 3.7854 liters are the same amount of volume, 1 gallon/3.7854 liters is 1. So suppose that you have measured the flow of water through a pipe as 283 liters/second. You can convert to gallons/minute by multiplying 283 liters/second by 1 twice: (283 liters/second)(60 seconds/1 minute)(1 gallon/3.7854 liters). When you complete this multiplication, the liter units cancel, the second units cancel, and you're left with 4,486 gallons/minute.
At low speeds, mass and energy appear to be separate quantities. Mass is the measure of inertia and can be determined by shaking an object. Energy is the measure of how much work an object can do and can be determined by letting it do that work. Conveniently enough, the object's weight—the force gravity exerts on it—is exactly proportional to its mass, which is why people carelessly interchange the words "mass" and "weight," even though they mean different things.
But when something is moving at speeds approaching the speed of light, mass and kinetic energy no longer separate so easily. In fact, the relativistic equations of motion are more complicated than those describing slow objects and the way in which gravity affects fast objects is more complicated than simply giving them "weight."
Overall, you can view the bending of light by gravity in one of two ways. First, you can view it approximately as gravity affecting not on mass, but also energy so that light falls because its energy gives it something equivalent to a "weight." Second, you can view it more accurately as the bending of light as caused by a change in the shape of space and time around a gravitating object. Space is curved, so that light doesn't travel straight as it moves past gravitating objects—it follows the curves of space itself. The second or Einsteinian view, which correctly predicts twice as much bending of light as the first or Newtonian view, is a little disconcerting. That's why it took some time for the theory of general relativity to be widely accepted. (Thanks to DP for pointing out the factor of two.)
These purported gravitational anomalies are just illusions. Because gravity is a relatively weak force, enormous concentrations of mass are required to create significant gravitational fields. Since it takes the entire earth to give you your normal weight, the mass concentration needed to cancel or oppose the earth's gravitation field in only one location would have to be extraordinary. While objects capable of causing such bizarre effects do exist elsewhere in our universe (e.g. black holes and neutron stars), there fortunately aren't any around here. As a result, the strength of the gravitational field at the earth's surface varies less than 1% over the earth's surface and always points almost exactly toward the center of the earth. Any tourist attraction that claims to have gravity pointing in some other direction with some other strength is claiming the impossible.
The fact that more massive objects also weigh more is just an observation of how the universe works. However, any other behavior would lead to some weird consequences. Suppose, for example, that an object's weight didn't depend on its mass, that all objects had the same weight. Then two separate balls would each weigh this standard amount. But now suppose that you glued the two balls together. If you think of them as two separate balls that are now attached, they should weigh twice the standard amount. But if you think of them as one oddly shaped object, they should weigh just the standard amount. Something wouldn't be right. So the fact that weight is proportional to mass is a sensible situation and also the way the universe actually works.
A ball bounces because its surface is elastic and it stores energy during the brief period of collision when the ball and floor are pushing very hard against one another. Much of this stored energy is released in a rebound that tosses the ball back upward for another bounce. But people don't store energy well during a collision and they don't rebound much. The energy that we should store is instead converted into thermal energy—we get hot rather than bouncing back upward.
The force that gravity exerts on an object is that object's weight. An object that has more gravity pulling on it weighs more and vice versa.
While you are throwing the ball upward, you are pushing it upward and there is an upward force on the ball. But as soon as the ball leaves your hand, that upward force vanishes and the ball travels upward due to its inertia alone. In the discussion of that upward flight, I always said "after the ball leaves your hand," to exclude the time when you are pushing upward on the ball. Starting and stopping demonstrations are often tricky and I meant you to pay attention only to the period when the ball was in free fall.
Copyright 1997-2017 © Louis A. Bloomfield, All Rights Reserved