by Walter Lewin
Now back to the case of the straw with your finger on top. If you raise the straw slowly up, say 5 centimeters, or about 2 inches, as long as the straw is still in the juice, the juice will not run out of the straw. In fact it will almost (not quite) stay exactly at the mark where it was before. You can test this by marking the side of the straw at the juice line before you lift it. The surface of the juice inside the straw will now be about 5 centimeters higher than the surface of the juice in the glass.
But given our earlier sacred statement about the pressure equalizing inside and outside of the straw—at the same level—how can this be? Doesn’t this violate the rule? No it does not! Nature is very clever; the air trapped by your finger in the straw will increase its volume just enough so that its pressure will decrease just the right amount (about 0.005 atmosphere) so that the pressure in the liquid in the straw at the same level of the surface of the liquid in the glass becomes the same: 1 atmosphere. This is why the juice will not rise precisely 5 centimeters, but rather just a little less, maybe only 1 millimeter less—just enough to give the air enough extra volume to lower its pressure to the desired amount.
Can you guess how high water (at sea level) can go in a tube when you’ve closed off one end and you slowly raise the tube upward? It depends on how much air was trapped inside the tube when you started raising it. If there was very little air in the straw, or even better no air at all, the maximum height the water could go would be about 34 feet—a little more than 10 meters. Of course, you couldn’t do this with a small glass, but a bucket of water might do. Does this surprise you? What makes it even more difficult to grasp is that the shape of the tube doesn’t matter. You could make it twist and even turn it into a spiral, and the water can still reach a vertical height of 34 feet, because 34 feet of water produces a hydrostatic pressure of 1 atmosphere.
Knowing that the lower the atmospheric pressure, the lower the maximum possible column of water will be, provides us with a way to measure atmospheric pressure. To see this, we could drive to the top of Mount Washington (about 6,300 feet high), where the atmospheric pressure is about 0.82 atmosphere, so this means that the pressure at the surface outside the tube is no longer 1 atmosphere but only about 0.82 atmosphere. So, when I measure the pressure in the water inside the tube at the level of the water surface outside the tube, it must also be 0.82 atmosphere, and thus the maximum possible height of the water column will be lower. The maximum height of water in the tube would then be 0.82 times 34 feet, which is about 28 feet.
If we measure the height of that column using cranberry juice by marking meters and centimeters on the tube, we have created a cranberry juice barometer—which will indicate changes in air pressure. The French scientist Blaise Pascal, by the way, is said to have made a barometer using red wine, which is perhaps to be expected of a Frenchman. The man credited with inventing the barometer in the mid-seventeenth century, the Italian Evangelista Torricelli, who was briefly an assistant to Galileo, settled eventually on mercury for his barometer. This is because, for a given column, denser liquids produce more hydrostatic pressure and so they have to rise less in the tube. About 13.6 times denser than water, mercury made the length of the tube much more convenient. The hydrostatic pressure of a 34-foot column of water (which is 1 atmosphere) is the same as 34 feet divided by 13.6 which is 2.5 feet of mercury (2.5 feet is 30 inches or 76 centimeters).
Torricelli wasn’t actually trying to measure air pressure at first with his device. He was trying to find out whether there was a limit to how high suction pumps could draw up a column of water—a serious problem in irrigation. He poured mercury to the top of a glass tube about 1 meter long, closed at the bottom. He then sealed the opening at the rim with his thumb and turned it upside down, into a bowl of mercury, taking his thumb away. When he did this, some of the mercury ran out of the tube back into the bowl, but the remaining column was about 76 centimeters high. The empty space at the top of the tube, he argued, was a vacuum, one of the very first vacuums produced in a laboratory. He knew that mercury was about 13.6 times denser than water, so he could calculate that the maximum length of a water column—which was what he really wanted to know—would be about 34 feet. While he was working this out, as a side benefit, he noticed that the level of the liquid rose and fell over time, and he came to believe that these changes were due to changes in atmospheric pressure. Quite brilliant. And his experiment explains why mercury barometers always have a little extra vacuum space at the top of their tubes.
Pressure Under Water
By figuring out the maximum height of a column of water, Torricelli also figured out something you may have thought about while trying to catch a glimpse of fish in the ocean. My hunch is you’ve probably tried snorkeling at some point in your life. Well, most snorkels have tubes no more than a foot long; I’m sure you’ve wanted to go deeper at times and wished the snorkel were longer. How deep do you think you could go and still have the snorkel work? Five feet, ten feet, twenty?
I like to find the answer to this question in class with a simple device called a manometer; it’s a common piece of lab equipment. It’s very simple, and you could easily make one at home, as I’ll describe in just a bit. What I really want to find out is how deep I can be below the surface and still suck air into my lungs. In order to figure this out, we have to measure the hydrostatic pressure of the water bearing in on my chest, which gets more powerful the deeper I go.
The pressure surrounding us, which is, remember, identical at identical levels, is the sum of the atmospheric pressure and the hydrostatic pressure. If I snorkel below the surface of the water, I breathe in air from the outside. That air has a pressure of 1 atmosphere. As a result, when I take air in through the snorkel, the pressure of the air in my lungs becomes the same, 1 atmosphere. But the pressure on my chest is the atmospheric pressure plus the hydrostatic pressure. So now the pressure on my chest is higher than the pressure inside my lungs; the difference is exactly the hydrostatic pressure. This causes no problem in exhaling, but when I inhale, I have to expand my chest. And if the hydrostatic pressure is too high because I’m too deep in the water, I simply don’t have the muscular strength to overcome the pressure difference, and I can’t take in more air. That’s why, if I want to go deeper in the water, I need to breathe in pressurized air to overcome the hydrostatic pressure. But highly pressurized air is quite taxing on our bodies, which is why there are strict limits to the amount of time for dives.
Now to come back to snorkeling, how far down can I go? To figure this out, I rig a manometer up on the wall of the lecture hall. Imagine a transparent plastic tube about 4 meters long. I attach one end to the wall high up on the left and then snake it into a U shape on the wall. Each arm of the U is a little less than 2 meters long. I pour about 2 meters’ worth of cranberry juice into the tube and it naturally settles to the same level on each side of the U tube. Now, by blowing into the right end of the tube I push the cranberry juice up on the left side of the U tube. The vertical distance I can push the juice up will tell me how deep I will be able to snorkel. Why? Because this is a measure of how much pressure my lungs can apply to overcome the hydrostatic pressure of the water—cranberry juice and water being for this purpose equivalent—but the cranberry juice is easier to see for the students.
I lean over, exhale completely, inhale to fill my lungs, take the right end of the tube in my mouth, and blow into it as hard as I can. My cheeks sink in, my eyes bug out, and the juice inches up in the left side of the U tube, and just barely rises by—could you guess?—50 centimeters. It takes everything I have to get it there, and I can’t hold it for more than a few seconds. So, I have pushed the juice up 50 centimeters on the left side, which means that I have also pushed it down 50 centimeters on the right side—in total, I have displaced the column of juice about 100 centimeters vertically, or one full meter (39 inches). Of course we are sucking air in when we breathe through a snorkel, not blowing it out. So perhaps it’s easier to suck the air in? So,
I do the experiment again, but this time I suck in the juice as far up the tube as I can. The result, however, is roughly the same; it only rises about 50 centimeters on the side that I suck—thus it goes down 50 centimeters on the other side, and I am utterly exhausted.
I have just imitated snorkeling 1 meter under water, the equivalent of one-tenth of an atmosphere. My students are invariably surprised by the demonstration, and they figure they can do better than their aging professor. So I invite a big strong guy to come up and give it a try, and after his best effort, his face is bright red, and he’s shocked. He’s only been able to do a little bit better—a couple of centimeters more—than I could.
This, it turns out, is just about the upper limit of how far down we can go and still breathe through a snorkel—1 lousy meter (about 3 feet). And we could really only manage this for a few seconds. That’s why most snorkels are much shorter than 1 meter, usually only about a foot long. Try making yourself a longer snorkel—you can do so with any kind of tubing—and see what happens.
You may wonder just how much force is exerted on your chest when you submerge to do a little snorkeling. At 1 meter below the surface, the hydrostatic pressure amounts to about one-tenth of an atmosphere, or we could say one-tenth of a kilogram per square centimeter. Now the surface area of your chest is roughly one square foot, about 1,000 square centimeters. Thus the force on your chest is about 1,100 kilograms, and the force on the inner wall of your chest due to the air pressure in your lungs is about 1,000 kilograms. Therefore the one-tenth of pressure difference translates into a difference in force of 100 kilograms—a 200-pound weight! When you look at it from this perspective, snorkeling looks a lot harder, right? And if you went down 10 meters, the hydrostatic pressure would be 1 full atmosphere, 1 kilogram per square centimeter of surface, and the force on your poor chest would be about 1,000 kilograms (1 ton) higher than the outward force produced by the 1-atmosphere pressure in your lungs.
This is why Asian pearl divers—some of whom routinely dove down 30 meters—risked their lives at such depths. Because they could not snorkel, they had to hold their breath, which they could do only for a few minutes, so they had to be quick about their work.
Only now can you really appreciate the engineering achievement represented by a submarine. Let’s think about a submarine at 10 meters down and assume that the air pressure inside is 1 atmosphere. The hydrostatic pressure (which is the pressure difference between outside and inside the sub) is about 10,000 kilograms per square meter, about 10 tons per square meter, so you can see that even a very small submarine has to be very strong to dive only 10 meters.
This is what makes the accomplishment of the fellow who invented the submarine in the early seventeenth century—Cornelis van Drebbel, who was Dutch, I’m happy to say—so astonishing. He could only operate it about 5 meters below the surface of the water, but even so, he had to deal with a hydrostatic pressure of half an atmosphere, and he built it of leather and wood! Accounts from the time say that he successfully maneuvered one of his crafts at this depth in trials on the Thames River, in England. This model was said to be powered by six oarsmen, could carry sixteen passengers, and could stay submerged for several hours. Floats held the “snorkels” just above the surface of the water. The inventor was hoping to impress King James I, trying to entice him to order a number of the crafts for his navy, but alas, the king and his admirals were not sufficiently impressed, and the sub was never used in combat. As a secret weapon, perhaps, van Drebbel’s sub was underwhelming, but as a feat of engineering it was absolutely remarkable. You can find out more about Van Drebbel and early submarines at this website: www.dutch submarines.com/specials/special_drebbel.htm.
Just how far down modern navy submarines can dive is a military secret, but the prevailing wisdom is that they can go about 1,000 meters (3300 feet) deep, where the hydrostatic pressure is around 100 atmospheres, a million kilos (1,000 tons) per square meter. Not surprisingly, U.S. subs are made of very high grade steel. Russian submarines are said to be able to go even deeper, because they’re made of stronger titanium.
It’s easy to demonstrate what would happen to a submarine if its walls weren’t strong enough, or if it dove too deep. To do this I hook up a vacuum pump to a gallon-size paint can and slowly pump the air out of the can. The pressure difference between the air outside and inside can only be as high as 1 atmosphere (compare that with the submarine!). We know that paint cans are fairly strong, but right before our eyes, because of the pressure difference, this one crumples like a flimsy aluminum soda can. It appears as though an invisible giant has taken hold of it and squeezed it in its fist. We’ve probably all done essentially the same thing at some point with a plastic water bottle, sucking a good bit of the air out of it and making it crumple. Intuitively, you may think the bottle scrunches up because of the power with which you’ve sucked on the bottle. But the real reason is that when I empty the air from the paint can, or you suck some of the air out of the water bottle, the outside air pressure no longer has enough competing pressure to push back against it. That’s what the pressure of our own atmosphere is ready to do at any moment. Absolutely any moment.
A metal paint can, a plastic water bottle—these are totally mundane things, right? But if we look at them the way a physicist does, we see something entirely different: a balance of fantastically powerful forces. Our lives would not be possible without these balances of largely invisible forces, forces due to atmospheric and hydrostatic pressure, and the inexorable force of gravity. These forces are so powerful that if—or when—they get even a little bit out of equilibrium, they can cause catastrophe. Suppose a leak develops in the seam of an airplane fuselage at 35,000 feet (where the atmospheric pressure is only about 0.25 atmospheres) while the plane is traveling at 550 miles per hour? Or a hairline crack opens up in the roof of the Baltimore Harbor Tunnel, 50 feet to 100 feet below the surface of the Patapsco River?
The next time you walk down a city street, try thinking like a physicist. What are you really seeing? For one, you are seeing the result of a furious battle raging inside every single building, and I don’t mean office politics. On one side of the battlefield, the force of Earth’s gravitational attraction is striving to pull all of it down—not only the walls and floors and ceilings, but the desks, air-conditioning ducts, mail chutes, elevators, secretaries and CEOs alike, even the morning coffee and croissants. On the other side, the combined force of the steel and brick and concrete and ultimately the ground itself are pushing the building up into the sky.
One way to think of architecture and construction engineering, then, is that they are the arts of battling the downward force to a standstill. We may think of certain feathery skyscrapers as having escaped gravity. They’ve done no such thing—they’ve taken the battle literally to new heights. If you think about it for a little while, you’ll see that the stalemate is only temporary. Building materials corrode, weaken, and decay, while the forces of our natural world are relentless. It’s only a matter of time.
These balancing acts may be most threatening in big cities. Consider a horrible accident that happened in New York City in 2007, when an eighty-three-year-old 2-foot-wide pipe beneath the street suddenly could no longer contain the highly pressurized steam it carried. The resulting geyser blew a 20-foot hole in Lexington Avenue, engulfing a tow truck, and shot up higher than the nearby seventy-seven-story Chrysler Building. If such potentially destructive forces were not held in exquisite balance nearly all of the time, no one would walk any city streets.
These stalemates between immensely powerful forces are not all the product of human handiwork. Consider trees. Calm, silent, immobile, slow, uncomplaining—they employ dozens of biological strategies to combat the force of gravity as well as hydrostatic pressure. What an achievement to sprout new branches every year, to continue putting new rings on its trunk, making the tree stronger even as the gravitational attraction between the tree and the earth grows more powerful. And still a tree pushes sap
up into its very highest branches. Isn’t it amazing that trees can be taller than about 10 meters? After all, water can only rise 10 meters in my straw, never higher; why (and how) would water be able to rise much higher in trees? The tallest redwoods are more than 300 feet tall, and somehow they pull water all the way up to their topmost leaves.
This is why I feel such sympathy for a great tree broken after a storm. Fierce winds, or ice and heavy snow accumulating on its branches, have managed to upset the delicate balance of forces the tree had been orchestrating. Thinking about this unending battle, I find myself appreciating all the more that ancient day when our ancestors stood on two legs rather than four and began to rise to the occasion.
Bernoulli and Beyond
There may be no more awe-inspiring human achievement in defying the incessant pull of gravity and mastering the shifting winds of air pressure than flight. How does it work? You may have heard that it has to do with Bernoulli’s principle and air flowing under and over the wings. This principle is named for the mathematician Daniel Bernoulli who published what we now call Bernoulli’s equation in his book Hydrodynamica in 1738. Simply put, the principle says that for liquid and gas flows, when the speed of a flow increases, the pressure in the flow decreases. That is hard to wrap your mind around, but you can see this in action.
