For the Love of Physics (8 page)

For all the galaxies whose redshifts and distance he could measure, Hubble found that the farther away these objects were, the faster they were moving away. So the universe was expanding. What a monumental discovery! Every galaxy in the universe speeding away from every other galaxy.

This can cause great confusion in the meaning of distance when galaxies are billions of light-years away. Do we mean the distance when the light was emitted (13 billion years ago, for instance) or do we mean the distance we think it is now, since the object has substantially increased its distance from us in those 13 billion years? One astronomer may report that the distance is about 13 billion light-years (this is called the light travel time distance) whereas another may report 29 billion light-years for the same object (this is called the co-moving distance).

Hubble’s findings have since become known as Hubble’s law: the velocity at which galaxies move away from us is directly proportional to their distance from us. The farther away a galaxy is, the faster it is racing away.

Measuring the velocities of the galaxies was relatively easy; the amount of redshift immediately translates into the speed of the galaxy. However, to get accurate distances was a different matter. That was the hardest part. Remember, Hubble’s distance to the Andromeda Nebula was off by a factor of 2.5. He came up with the fairly simple equation
v
=
H
₀
D
, where
v
is the velocity of a given galaxy,
D
is the distance of that galaxy from us, and
H
₀
is a constant, now called Hubble’s constant. Hubble estimated the constant to be about 500, measured in units of kilometers per second per megaparsec (1 megaparsec is 3.26 million light-years). The uncertainty in his constant was about 10 percent. Thus, as an example, according to Hubble, if a galaxy is at a distance of 5 megaparsecs, its speed relative to us is about 2,500 kilometers per second (about 1,600 miles per second).

Clearly the universe is expanding fast. But that wasn’t all Hubble’s discovery revealed. If you really knew the value of Hubble’s constant, then you could turn the clock backward in order to calculate the time since
the big bang, and thus the age of the universe. Hubble himself estimated that the universe was about 2 billion years old. This calculation was in conflict with the age of the Earth, which geologists were just measuring to be upward of 3 billion years. This bothered Hubble mightily, for good reason. Of course, he was unaware of a number of systematic errors he was making. Not only was he confusing different kinds of Cepheid variables in some cases, but he also mistook clouds of gas in which stars were forming for bright stars in faraway galaxies.

One way of looking at eighty years’ worth of progress in measuring stellar distances is to look at the history of Hubble’s constant itself. Astronomers have been struggling to nail down the value of Hubble’s constant for nearly a century, which has produced not only a seven-fold reduction in the constant, which dramatically increased the size of the universe, but also changed the age of the universe, from Hubble’s original 2 billion years to our current estimate of nearly 14 billion years—actually 13.75 ± 0.11 billion years. Now, finally, based on observations in part from the fabulous orbiting telescope bearing Hubble’s name, we have a consensus that Hubble’s constant is 70.4 ± 1.4 kilometers per second per megaparsec. The uncertainty is only 2 percent—which is incredible!

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

H
ere’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.