by Walter Lewin
Hold a single sheet of paper, say an 8.5 × 11–inch standard sheet, up to your mouth (not in your mouth) with the short edge at your mouth. The paper will curl down because of gravity. Now blow hard straight out across the top of the paper, and watch what happens. You’ll see the paper rise. And depending on how hard you blow, you can make the paper really jump up. You’ve just demonstrated Bernoulli’s principle, and this simple phenomenon also helps explain how airplanes fly. Though many of us may have become used to the sight, watching a 747 take off, or being strapped in a seat when the thing lifts off, is a truly strange experience. Just watch the delight with which little children see their first plane take off. A Boeing 747-8 has a maximum takeoff weight of nearly a million pounds. How on earth does it stay aloft?
An airplane wing is designed so that the air that passes above it speeds up relative to the air that passes underneath it. Because of Bernoulli, the faster airflow on top of the wing lowers the air pressure above the wing, and the resulting difference between that low pressure and the higher pressure under the wing provides upward lift. Let’s call this Bernoulli lift. Many physics books tell you that Bernoulli lift is entirely responsible for the upward lift of airplanes—in fact, this idea is all over the place. And yet, if you think about it for a minute or two, you can figure out that it cannot be true. Because if it were true, how would planes ever fly upside down?
So it’s obvious that Bernoulli’s principle alone cannot be the sole reason for the upward lift. In addition to the Bernoulli lift there is a so-called reaction lift. B. C. Johnson describes this in detail in his delightful article “Aerodynamic Lift, Bernoulli Effect, Reaction Lift” (http://mb-soft.com/public2/lift.html). Reaction lift (named for Newton’s third law: for every action there is an equal and opposite reaction) comes about when air passes underneath an airplane wing angled upward. That air, moving from the front of the wing to the back, is pushed downward by the wing. That’s the “action.” That action must be met by an equal reaction of air pushing upward, so there is upward lift on the wing. In the case of a Boeing 747 (cruising at 550 miles per hour at an altitude of about 30,000 feet) more than 80 percent of the lift comes from reaction lift, and less than 20 percent from Bernoulli lift.
You can demonstrate reaction lift pretty easily yourself the next time you travel in a car. In fact, you may even have done this when you were little. When the car is moving, roll down the window, stick your arm outside, hold your hand in the direction that the car is moving, and tilt the angle of your hand such that your fingers are pointing upward. You will feel your hand pushed upward. Voila! Reaction lift.
You may think now that you understand why some planes can fly upside down. However, do you realize that if a plane rolls over 180 degrees that both the Bernoulli force and the reaction force will now be pointing downward? Remember, in normal flight the reaction force is upward because the wings are angled upward, but after a 180-degree rollover, they will be angled downward.
Do the experiment again to feel the reaction lift on your hand. As long as you tilt your fingers upward you will feel an upward force. Now change the angle such that your fingers are tilted downward; you will now feel a force in the downward direction.
Why then is it possible to fly upside down? The required lift must somehow come from an upward reaction force, since that’s the only game in town. This becomes possible if the pilot (flying upside down) raises the front end of the plane enough so that the wings become angled upward again. This is a tricky business and only very experienced pilots can do it. It’s also rather dangerous to rely solely on reaction lift, since by nature reaction lift is not very stable. You can sense this instability doing the experiment with your hand outside the car window. Your hand jiggles around quite a bit. In fact, it’s this difficulty in controlling reaction lift that accounts for why most airplane crashes occur close to takeoff and landing. The fraction of lift accounted for by reaction lift is higher at takeoff and landing than during flight at normal altitude. This is why when a big airliner lands, you can sometimes feel the plane wobble.
The Drink Thief
The mysteries of pressure are in truth almost endlessly perplexing. Come back, for example, to the physics of drinking with a straw. Here is one last puzzle to consider, a wonderful brainteaser.
At home one weekend I said to myself, “I wonder what would be the longest straw that I could drink a glass of juice from.” We’ve all seen super-long straws, often with turns and twists in them, which children love.
We saw earlier that we can only suck hard enough to displace juice about a maximum of 1 meter—and that only for a few seconds—meaning that I would not be able to suck up juice with a straw any higher than 1 meter (about 3 feet). So I decided to cut myself a piece of thin plastic tube 1 meter long and see if that would work. No problem; I could suck the juice up just fine. So I decided to cut a piece 3 meters long—that’s almost 10 feet—and I got up on a chair in my kitchen and put a bucket of water on the floor, and sure enough, I could suck it up that far too. Amazing. Then I thought to myself, if I were up on the second story of my house and I looked down at someone below, say out on a deck having a great big tumbler of juice, wine, or whatever—let’s say a very large cranberry and vodka—could I steal that drink by sucking it up if I had a really long straw? I decided to find out, and this led to one of the demonstrations I love to do in class. It never ceases to amaze the students.
I pull out a long length of coiled-up clear plastic tubing and I ask for a front-row volunteer. I place a large glass beaker of cranberry juice—no vodka—on the floor in the classroom for all students to see. Holding the tubing, I begin to climb a tall ladder; it reaches about 16 feet off the floor—almost 5 meters!
“Okay, here’s my straw,” I say, dropping one end of the tubing to the student. She holds the end in the beaker, and I can feel the students’ anticipation. The class can’t quite believe I’m up there. Remember, they were witnesses to the fact that I could only displace the cranberry juice about 1 meter, or about 3 feet. Now I’m about 16 feet off the ground. How could I possibly do it?
I begin sucking, grunting a bit as the juice rises slowly inside the tube: first 1 meter, then 2, and then 3. Then the level dips a little, but soon the juice resumes climbing very slowly again until it reaches my mouth. I say a loud “Mmmmm” and the class erupts in applause. What has been going on here? Why could I suck the juice up so high?
Frankly, I cheat. Not that it matters, since there are no rules in the game. Every time after sucking, when I can’t take any more air in, I put my tongue over the end of the tube. In other words I close the tube off, and as we saw earlier, this will keep the juice up in the tube. I then exhale and I start sucking again, and repeat that scenario many times. My mouth becomes a kind of suction pump and my tongue a kind of stop valve.
To make the juice rise those 16 feet, I have to lower the pressure of the air in the tube to about half an atmosphere. And yes, if you’re wondering, I could have used the same trick with the manometer, and I would have been able to suck up a much longer column of cranberry juice. Does that mean that I could also snorkel much farther down beneath the surface of a lake or the sea?
What do you think? If you know the answer, drop me a note!
CHAPTER 5
Over and Under—Outside and Inside—the Rainbow
So many of the little wonders of the everyday world—which can be truly spectacular—go unobserved most of the time because we haven’t been trained how to see them. I remember one morning, four or five years ago, when I was drinking my morning espresso in my favorite red and blue Rietveld chair, and suddenly I noticed this beautiful pattern of round spots of light on the wall, amidst the flickering of shadows thrown by the leaves of a tree outside the window. I was so delighted to have spotted them that my eyes lit up. Not sure what had happened, but with her usual astuteness, my wife, Susan, wondered if something was the matter.
“Do you know what that is?” I responded, pointing to the
light circles. “Do you understand why that’s happening?” Then I explained. You might expect the light to make lots of little shimmerings on the wall rather than circles, right? But each of the many small openings between the leaves was acting like a camera obscura, a pinhole camera, and such a camera reproduces the image of the light source—in this case the Sun. No matter what the shapes of the openings through which the light is streaming, as long as the opening is small, it’s the shape of the light source itself that’s re-created on the wall.
So during a partial solar eclipse, sunlight pouring through my window wouldn’t make circles on my wall anymore—all the circles would have a bite taken out of them, because that would be the shape of the Sun. Aristotle knew this more than two thousand years ago! It was fantastic to see those light spots, right there on my bedroom wall, demonstrating the remarkable properties of light.
Secrets of the Rainbow
In truth, the marvelous effects of the physics of light are everywhere we look, sometimes in the most ordinary sights, and at other times in some of nature’s most beautiful creations. Take rainbows, for example: fantastic, wonderful phenomena. And they’re everywhere. Great scientists—Ibn al-Haytham, the eleventh-century Muslim scientist and mathematician known as the father of optics, the French philosopher, mathematician, and physicist René Descartes; and Sir Isaac Newton himself—found them captivating and tried to explain them. Yet most physics teachers ignore rainbows in their classes. I can’t believe this; in fact, I think it’s criminal.
Not that the physics of rainbows is simple. But so what? How can we refuse to tackle something that pulls so powerfully on our imaginations? How can we not want to understand the mystery behind the intrinsic beauty of these glorious creations? I have always loved lecturing about them, and I tell my students, “At the end of this lecture, your life will never be the same again, never.” The same goes for you.
Former students and others who’ve watched my lectures on the web have been mailing and emailing me wonderful images of rainbows and other atmospheric phenomena for decades. I feel as though I have a network of rainbow scouts spread across the world. Some of these shots are extraordinary—especially those from Niagara Falls, which has so much spray that the bows are spectacular. Maybe you will want to send me pictures too. Feel free!
I’m sure you’ve seen at least dozens, if not hundreds, of rainbows in your life. If you’ve spent time in Florida or Hawaii, or other tropical locations where there are frequent rain showers while the Sun shines, you’ve seen even more. If you’ve watered your garden with a hose or sprinkler when the Sun is shining, you’ve probably created rainbows.
Most of us have looked at many rainbows, yet very few of us have ever seen rainbows. Ancient mythologies have called them gods’ bows, bridges or paths between the homes of mortals and the gods. Most famously in the West, the rainbow represented God’s promise in the Hebrew Bible never again to bring an all-destroying flood to the earth: “I do set my bow in the clouds.”
Part of the charm of rainbows is that they are so expansive, spreading majestically, and so ephemerally, across the entire sky. But, as is so often true in physics, their origins lie in extraordinarily large numbers of something exceptionally minute: tiny spherical balls of water, sometimes less than 1 millimeter (1/25 of an inch) across, floating in the sky.
While scientists have been trying to explain the origins of rainbows for at least a millennium, it was Isaac Newton who offered the first truly convincing explanation in his 1704 work Opticks. Newton understood several things at once, all of which are essential for producing rainbows. First, he demonstrated that normal white light was composed of all the colors (I was going to say of “all the colors of the rainbow,” but that would be getting ahead of ourselves). By refracting (bending) light through a glass prism, he separated it into its component colors. Then, by sending the refracted light back through another prism, he combined the colored light back into white light, proving that the prism itself wasn’t creating the colors in some way. He also figured out that many different materials could refract light, including water. And this is how he came to understand that raindrops refracting and reflecting light were the key to producing a rainbow.
A rainbow in the sky, Newton concluded correctly, is a successful collaboration between the Sun, zillions of raindrops, and your eyes, which must be observing those raindrops at just the right angles. In order to understand just how a rainbow is produced, we need to zero in on what happens when light enters a raindrop. But remember, everything I’m going to say about this single raindrop in reality applies to the countless drops that make up the rainbow.
For you to see a rainbow, three conditions need to be met. First, the Sun needs to be behind you. Second, there must be raindrops in the sky in front of you—this could be miles or just a few hundred yards away. Third, the sunlight must be able to reach the raindrops without any obstruction, such as clouds.
When a ray of light enters a raindrop and refracts, it separates into all of its component colors. Red light refracts, or bends, the least, while violet light refracts the most. All of these different-colored rays continue traveling toward the back of the raindrop. Some of the light keeps going and exits the raindrop, but some of it bounces back, or reflects, at an angle, toward the front of the raindrop. In fact, some of the light reflects more than once, but that only becomes important later. For the time being, we are only interested in the light that reflects just once. When the light exits the front of the drop, some of the light again refracts, separating the different colored rays still further.
After these rays of sunlight refract, reflect, and refract again on their way out of the raindrop, they have pretty much reversed direction. Key to why we see rainbows is that red light exits the raindrop at angles that are always smaller than about 42 degrees from the original direction of the sunlight entering the drop. And this is the same for all raindrops, because the Sun for all practical purposes is infinitely far away. The angle at which the red light exits can be anything between 0 degrees and 42 degrees but never more than 42 degrees, and this maximum angle is different for each of the different colors. For violet light, the maximum angle is about 40 degrees. These different maximum angles for each color account for the stripes of colors in the rainbow.
There is an easy way to spot a rainbow when conditions are right. As seen in the following figure, if I trace a line from the Sun through my head to the far end of my shadow on the ground, that line is precisely parallel to the direction from the Sun to the raindrops. The higher the Sun in the sky, the steeper this line will be, and the shorter my shadow. The converse is also the case. This line, from the Sun, through my head, to the shadow of my head on the ground, we will call the imaginary line. This line is very important as it will tell you where in the sky you should look to see the rainbow.
All raindrops at 42 degrees from the “imaginary line” will be red. Those at 40 degrees will be blue. raindrops at angles smaller than 40 degrees will be white (like the sunlight). We will see no light from drops at angles larger than 42 degrees (see text).
If you look about 42 degrees away from that imaginary line—it doesn’t matter whether it’s straight up, to the right, or to the left—that’s where you will see the red band of the rainbow. At about 40 degrees away from it—up, right, or left—you will see the violet band of the rainbow. But the truth is that violet is hard to see in a rainbow, so you’ll see the blue much more easily. Therefore we’ll just say blue from now on. Aren’t these the same angles I mentioned earlier, talking about the maximum angles of the light leaving the raindrop? Yes, and it’s no accident. Look again at the figure.
What about the blue band in the rainbow? Well, remember its magic number is just about 40 degrees, 2 degrees less than the red band. So blue light can be found refracting, reflecting, and refracting out of different raindrops at a maximum angle of 40 degrees. Thus we see blue light 40 degrees away from the imaginary line. Since the 40-degree band is closer to the imagin
ary line than the 42-degree band, the blue band will always be on the inside of the red band of the rainbow. The other colors making up the bow—orange, yellow, green—are found between the red and blue bands. For more about this you can take a look at my lecture on rainbows online, at http://ocw.mit.edu/courses/physics/8-03-physics-iii-vibrations-and-waves-fall-2004/video-lectures/lecture-22/.
Now you might wonder, at the maximum angle for blue light, are we seeing only blue light? After all, red light can also emerge at 40 degrees, as it is smaller than 42 degrees. If you’ve asked this question, more power to you—it’s a very astute one. The answer is that at the maximum angle for any given color, that color dominates all other colors. With red, though, because its angle is the highest, it is the only color.
Why is the rainbow a bow and not a straight line? Go back to that imaginary line from your eyes to the shadow of your head, and the magic number 42 degrees. When you measure 42 degrees—in all directions—away from the imaginary line, you are tracing an arc of color. But you know that not all rainbows are full arcs—some are just little pieces in the sky. That happens when there aren’t enough raindrops in all directions in the sky or when certain parts of the rainbow are in the shadow of obstructing clouds.
There’s another important aspect to this collaboration between the Sun, the raindrops, and your eyes, and once you see it, you’ll understand lots more about why rainbows—natural as well as artificial—are the way they are. For example, why are some rainbows enormous, while others just hug the horizon? And what accounts for the rainbows you sometimes see in pounding surf, or in fountains, waterfalls, or the spray of your garden hose?
Let’s go back to the imaginary line that runs from your eyes to the shadow of your head. This line starts at the Sun, behind you, and extends to the ground. However, in your mind, you can extend this line as far as you want, even much farther than the shadow of your head. This imaginary line is very useful, as you can imagine it going through the center (called the antisolar point) of a circle, on the circumference of which is the rainbow. This circle represents where the rainbow would form if the surface of Earth didn’t get in its way. Depending upon how high in the sky the Sun is, a rainbow will also be lower or higher above the horizon. When the Sun is very high, the rainbow may only just peek above the horizon, whereas late in the afternoon just before sunset, or early in the morning just around sunrise, when the Sun is low in the sky and when your shadow is long, then a rainbow may be enormous, reaching halfway up into the sky. Why halfway? Because the maximum angle it can be over the horizon is 42 degrees, or close to 45 degrees, which is half of the 90 degrees that would be right overhead.
