by Walter Lewin
Just think about it. Parallax measurements, starting in 1838, became the foundation for developing the instruments and mathematical tools to reach billions of light-years to the edge of the observable universe.
For all of our remarkable progress in solving mysteries such as this, there are of course a great many mysteries that remain. We can measure the proportion of dark matter and dark energy in the universe, but we have no idea what they are. We know the age of the universe but still wonder when or if and how it will end. We can make very precise measurements of gravitational attraction, electromagnetism, and of the weak and the strong nuclear forces, but we have no clue if they will ever be combined into one unified theory. Nor do we have any idea what the chances are of other intelligent life existing in our own or some other galaxy. So we have a long way to go. But the wonder is just how many answers the tools of physics have provided, to such a remarkably high degree of accuracy.
CHAPTER 3
Bodies in Motion
Here’s something fun to try. Stand on a bathroom scale—not one of those fancy ones at your doctor’s office, and not one of those digital glass things you have to tap with your toes to make it turn on, just an everyday bathroom scale. It doesn’t matter if you have your shoes on (you don’t have to impress anyone), and it doesn’t matter what number you see, and whether you like it or not. Now, quickly raise yourself up on your toes; then stop and hold yourself there. You’ll see that the scale goes a little crazy. You may have to do this several times to clearly see what’s going on because it all happens pretty quickly.
First the needle goes up, right? Then it goes way down before it comes back to your weight, where it was before you moved, though depending on your scale, the needle (or numbered disk) might still jiggle a bit before it stabilizes. Then, as you bring your heels down, especially if you do so quickly, the needle first goes down, then shoots up past your weight, before coming to rest back at the weight you may or may not have wanted to know. What was that all about? After all, you weigh the same whether you move your heels down or up on your toes, right? Or do you?
To figure this out, we need, believe it or not, Sir Isaac Newton, my candidate for the greatest physicist of all time. Some of my colleagues disagree, and you can certainly make a case for Albert Einstein, but no one really questions whether Einstein and Newton are the top two. Why do I vote for Newton? Because his discoveries were both so fundamental and so diverse. He studied the nature of light and developed a theory of color. To study the planetary motions he built the first reflecting telescope, which was a major advance over the refracting telescopes of his day, and even today almost all the major telescopes follow the basic principles of his design. In studying the properties of the motion of fluids, he pioneered a major area of physics, and he managed to calculate the speed of sound (he was only off by about 15 percent). Newton even invented a whole new branch of mathematics: calculus. Fortunately, we don’t need to resort to calculus to appreciate his most masterful achievements, which have come to be known as Newton’s laws. I hope that in this chapter I can show you how far-reaching these apparently simple laws really are.
Newton’s Three Laws of Motion
The first law holds that a body at rest will persist in its state of being at rest, and a body in motion will persist in its motion in the same direction with the same speed—unless, in either case, a force acts on it. Or, in Newton’s own words, “A body at rest perseveres in its state of rest, or of uniform motion in a right line unless it is compelled to change that state by forces impressed upon it.” This is the law of inertia.
The concept of inertia is familiar to us, but if you reflect on it for a bit, you can appreciate how counterintuitive it actually is. We take this law for granted now, even though it runs clearly against our daily experience. After all, things that move rarely do so along a straight line. And they certainly don’t usually keep moving indefinitely. We expect them to come to a stop at some point. No golfer could have come up with the law of inertia, since so few putts go in a straight line and so many stop well short of the hole. What was and still is intuitive is the contrary idea—that things naturally tend toward rest—which is why it had dominated Western thinking about these matters for thousands of years until Newton’s breakthrough.
Newton turned our understanding of the motion of objects on its head, explaining that the reason a golf ball often stops short of the hole is that the force of friction is slowing it down, and the reason the Moon doesn’t shoot off into space, but keeps circling Earth, is that the force of gravitational attraction is holding it in orbit.
To appreciate the reality of inertia more intuitively, think about how difficult it can be when you are ice skating to make the turn at the end of the rink—your body wants to keep going straight and you have to learn just how much force to apply to your skates at just the right angle to move yourself off of that course without flailing wildly or crashing into the wall. Or if you are a skier, think of how difficult it can be to change course quickly to avoid another skier hurtling into your path. The reason we notice inertia so much more in these cases than we generally do is that in both cases there is so little friction acting to slow us down and help us change our motion. Just imagine if putting greens were made of ice; then you would become acutely aware of just how much the golf ball wants to keep going and going.
Consider just how revolutionary an insight this was. Not only did it overturn all previous understanding; it pointed the way to the discovery of a host of forces that are acting on us all the time but are invisible—like friction, gravity, and the magnetic and electric forces. So important was his contribution that in physics the unit of force is called a newton. But not only did Newton allow us to “see” these hidden forces; he also showed us how to measure them.
With the second law he provided a remarkably simple but powerful guide for calculating forces. Considered by some the most important equation in all of physics, the second law is the famous F = ma. In words: the net force, F, on an object is the mass of the object, m, multiplied by the net acceleration, a, of the object.
To see just one way in which this formula is so useful in our daily lives, take the case of an X-ray machine. Figuring out how to produce just the right range of energies for the X-rays is crucial. Here’s how Newton’s equation lets us do just that.
One of the major findings in physics—which we’ll explore more later—is that a charged particle (say an electron or proton or ion) will experience a force when it is placed in an electric field. If we know the charge of the particle and the strength of the electric field, we can calculate the electric force acting on that particle. However, once we do know the force, using Newton’s second law we can calculate the acceleration of the particle.*
In an X-ray machine electrons are accelerated before they strike a target inside the X-ray tube. The speed with which the electrons hit the target determines the energy range of the X-rays that are then produced. By changing the strength of the electric field, we can change the acceleration of the electrons. Thus the speed with which the electrons hit the target can be controlled to select the desired energy range of the X-rays.
In order to facilitate making such calculations, physicists use as a unit of force, the newton—1 newton is the force that accelerates a mass of 1 kilogram at 1 meter per second per second. Why do we say “per second per second”? Because with acceleration, the velocity is constantly changing; so, in other words, it doesn’t stop after the first second. If the acceleration is constant, the velocity is changing by the same amount every second.
To see this more clearly, take the case of a bowling ball dropped from a tall building in Manhattan—why not from the observation deck of the Empire State Building? It is known that the acceleration of objects dropped on Earth is approximately 9.8 meters per second per second; it is called the gravitational acceleration, represented in physics by g. (For simplicity I am ignoring air drag for now; more about this later.) After the first second the bowling
ball has a speed of 9.8 meters per second. By the end of the second second, it will pick up an additional 9.8 meters per second of speed, so it will be moving at 19.6 meters per second. And by the end of the third second it will be traveling 29.4 meters per second. It takes about 8 seconds for the ball to hit the ground. Its speed is then about 8 times 9.8, which is about 78 meters per second (about 175 miles per hour).
What about the much repeated notion that if you threw a penny off the top of the Empire State Building it would kill someone? I’ll again exclude the role of air drag, which I emphasize would be considerable in this case. But even without that factored in, a penny hitting you with a speed of about 175 miles per hour will probably not kill you.
This is a good place to grapple with an issue that will come up over and over in this book, mainly because it comes up over and over in physics: the difference between mass and weight. Note that Newton used mass in his equation rather than weight, and though you might think of the two as being the same, they’re actually fundamentally different. We commonly use the pound and the kilogram (the units we’ll use in this book) as units of weight, but the truth is that they are units of mass.
The difference is actually simple. Your mass is the same no matter where you are in the universe. That’s right—on the Moon, in outer space, or on the surface of an asteroid. It’s your weight that varies. So what is weight, then? Here’s where things get a little tricky. Weight is the result of gravitational attraction. Weight is a force: it is mass times the gravitational acceleration (F = mg). So our weight varies depending upon the strength of gravity acting on us, which is why astronauts weigh less on the Moon. The Moon’s gravity is about a sixth as strong as Earth’s, so on the Moon astronauts weigh about one-sixth what they weigh on Earth.
For a given mass, the gravitational attraction of the Earth is about the same no matter where you are on it. So we can get away with saying, “She weighs a hundred twenty pounds”* or “He weighs eighty kilograms,”* even though by doing so we are confusing these two categories (mass and weight). I thought long and hard about whether to use the technical physics unit for force (thus weight) in this book instead of kilos and pounds, and decided against it on the grounds that it would be too confusing—no one, not even a physicist whose mass is 80 kilograms would say, “I weigh seven hundred eighty-four newtons” (80 × 9.8 = 784). So instead I’ll ask you to remember the distinction—and we’ll come back to it in just a little while, when we return to the mystery of why a scale goes crazy when we stand on our tiptoes on it.
The fact that gravitational acceleration is effectively the same everywhere on Earth is behind a mystery that you may well have heard of: that objects of different masses fall at the same speed. A famous story about Galileo, which was first told in an early biography, recounts that he performed an experiment from the top of the Leaning Tower of Pisa in which he threw a cannonball and a smaller wooden ball off the tower at the same time. His intent, reputedly, was to disprove an assertion attributed to Aristotle that heavier objects would fall faster than light ones. The account has long been doubted, and it seems pretty clear now that Galileo never did perform this experiment, but it still makes for a good story—such a good story that the commander of the Apollo 15 Moon mission, David Scott, famously dropped a hammer and a falcon feather onto the surface of the Moon at the same time to see if objects of different mass would fall to the ground at the same rate in a vacuum. It’s a wonderful video, which you can access here: http://video.google.com/videoplay?docid=6926891572259784994#.
The striking thing to me about this video is just how slowly they both drop. Without thinking about it, you might expect them both to drop quickly, at least surely the hammer. But they both fall slowly because the gravitational acceleration on the Moon is about six times less than it is on Earth.
Why was Galileo right that two objects of different mass would land at the same time? The reason is that the gravitational acceleration is the same for all objects. According to F = ma, the larger the mass, the larger the gravitational force, but the acceleration is the same for all objects. Thus they reach the ground with the same speed. Of course, the object with the larger mass will have more energy and will therefore have a greater impact.
Now it’s important to note here that the feather and the hammer would not land at the same time if you performed this experiment on Earth. This is the result of air drag, which we’ve discounted until now. Air drag is a force that opposes the motion of moving objects. Also wind would have much more effect on the feather than on the hammer.
This brings us to a very important feature of the second law. The word net in the equation as given above is vital, as nearly always in nature more than one force is acting on an object; all have to be taken into account. This means that the forces have to be added. Now, it’s not really as simple as this, because forces are what we call vectors, meaning that they have a magnitude as well as a direction, which means that you cannot really make a calculation like 2 + 3 = 5 for determining the net force. Suppose only two forces act on a mass of 4 kilograms; one force of 3 newtons is pointing upward, and another of 2 newtons is pointing downward. The sum of these two forces is then 1 newton in the upward direction and, according to Newton’s second law, the object will be accelerated upward with an acceleration of 0.25 meters per second per second.
The sum of two forces can even be zero. If I place an object of mass m on my table, according to Newton’s second law, the gravitational force on the object is then mg (mass × gravitational acceleration) newtons in the downward direction. Since the object is not being accelerated, the net force on the object must be zero. That means that there must be another force of mg newtons upward. That is the force with which the table pushes upward on the object. A force of mg down and one of mg up add up to a force of zero!
This brings us to Newton’s third law: “To every action there is always an equal and opposite reaction.” This means that the force that two objects exert on each other are always equal and are directed in opposite directions. As I like to put it, action equals minus reaction, or, as it’s known more popularly, “For every action there is an equal and opposite reaction.”
Some of the implications of this law are intuitive: a rifle recoils backward against your shoulder when it fires. But consider also that when you push against a wall, it pushes back on you in the opposite direction with the exact same force. The strawberry shortcake you had for your birthday pushed down on the cake plate, which pushed right back at it with an equal amount of force. In fact, odd as the third law is, we are completely surrounded by examples of it in action.
Have you ever turned on the faucet connected to a hose lying on the ground and seen the hose snake all over the place, maybe spraying your little brother if you were lucky? Why does that happen? Because as the water is pushed out of the hose, it also pushes back on the hose, and the result is that the hose is whipped all around. Or surely you’ve blown up a balloon and then let go of it to see it fly crazily around the room. What’s happening is that the balloon is pushing the air out, and the air coming out of the balloon pushes back on the balloon, making it zip around, an airborne version of the snaking garden hose. This is no different from the principle behind jet planes and rockets. They eject gas at a very high speed and that makes them move in the opposite direction.
Now, to truly grasp just how strange and profound an insight this is, consider what Newton’s laws tell us is happening if we throw an apple off the top of a thirty-story building. We know the acceleration will be g, about 9.8 meters per second per second. Now, say the apple is about half a kilogram (about 1.1 pounds) in mass. Using the second law, F = ma, we find that the Earth attracts the apple with a force of 0.5 × 9.8 = 4.9 newtons. So far so good.
But now consider what the third law demands: if the Earth attracts the apple with a force of 4.9 newtons, then the apple will attract the Earth with a force of 4.9 newtons. Thus, as the apple falls to Earth, the Earth falls to the apple. This seems ridicul
ous, right? But hold on. Since the mass of the Earth is so much greater than that of the apple, the numbers get pretty wild. Since we know that the mass of the Earth is about 6 × 1024 kilograms, we can calculate how far it falls up toward the apple: about 10–22 meters, about one ten-millionth of the size of a proton, a distance so small it cannot even be measured; in fact, it’s meaningless.
This whole idea, that the force between two bodies is both equal and in opposite directions, is at play everywhere in our lives, and it’s the key to why your scale goes berserk when you lift yourself up onto your toes on it. This brings us back to the issue of just what weight is, and lets us understand it more precisely.
When you stand on a bathroom scale, gravity is pulling down on you with force mg (where m is your mass) and the scale is pushing up on you with the same force so that the net force on you is zero. This force pushing up against you is what the scale actually measures, and this is what registers as your weight. Remember, weight is not the same thing as mass. For your mass to change, you’d have to go on a diet (or, of course, you might do the opposite, and eat more), but your weight can change much more readily.
Let’s say that your mass (m) is 55 kilograms (that’s about 120 pounds). When you stand on a scale in your bathroom, you push down on the scale with a force mg, and the scale will push back on you with the same force, mg. The net force on you is zero. The force with which the scale pushes back on you is what you will read on the scale. Since your scale may indicate your weight in pounds, it will read 120 pounds.
Let’s now weigh you in an elevator. While the elevator stands still (or while the elevator is moving at constant speed), you are not being accelerated (neither is the elevator) and the scale will indicate that you weigh 120 pounds, as was the case when you weighed yourself in your bathroom. We enter the elevator (the elevator is at rest), you go on the scale, and it reads 120 pounds. Now I press the button for the top floor, and the elevator briefly accelerates upward to get up to speed. Let’s assume that this acceleration is 2 meters per second per second and that it is constant. During the brief time that the elevator accelerates, the net force on you cannot be zero. According to Newton’s second, the net force Fnet on you must be Fnet = manet. Since the net acceleration is 2 meters per second per second, the net force on you is m × 2 upward. Since the force of gravity on you is mg down, there must be a force of mg + m2, which can also be written as m(g + 2), on you in upward direction. Where does this force come from? It must come from the scale (where else?). The scale is exerting a force m(g + 2) on you upward. But remember that the weight that the scale indicates is the force with which it pushes upward on you. Thus the scale tells you that your weight is about 144 pounds (remember, g is about 10 meters per second per second). You have gained quite a bit of weight!
