Newton's Law Passes the Test

[Essayist]

Photos.com
LFFT: Isaac Newton, the English mathematician whose equation F = ma lies at the
foundation of almost all of the physical sciences. RIGHT: Fritz Zwicky, the Swiss
astrophysist who first proposed that some unseen "dark matter" held galaxies together.

F = ma. In words: Force = mass times acceleration.

It's Isaac Newton's second law: one of the first equations students learn in high school physics. It lies at the basis of almost all of the physical sciences. Engineers depend on it every day to build pretty much anything that moves. And yet, a group of rogue physicists has been trying to discredit the second law for over 20 years—and has been building a strong case and a following.

But in a recent Physical Review paper, a team of physicists reported that it has tested Newton's second law at very low accelerations—precisely where the second-law opponents say that it breaks down—and the law holds. While it is not entirely conclusive, this simple experiment carries weighty implications for the basis of physics, dark matter, the elusive "theory of everything" and, well, the fate of the universe.

And you thought it was just a few letters with an equal sign between them?

A Dark Fudge Factor

University of California, Santa Cruz
Physicists now believe that dark matter makes up the vast majority of the matter in the
universe—and that there might be just enough to keep the universe expanding at its
current rate.

So called "dark matter" might be one of the most important questions in astronomy today. The more astrophysicists look at the universe, the less it seems to make sense—at least, if we're actually seeing everything that's out there.

When you swing a ball on a string around your head, you have to pull in the string—otherwise the ball flies off. The heavier the ball, and the faster the ball is going, the tighter you have to hold. You are providing what physicists call the "centripetal force"—the force that keeps an object traveling in a circle by pulling it in towards the center.

Galaxies, like the swinging ball, rotate in a circle. The force that holds them together, and keeps each star from flying off—the centripetal force for galaxies—is gravity. The stars in the center exert a gravitational force on the stars flying around the edges, and hold them in.

But in 1933, Swiss astrophysicist Fritz Zwicky noticed that the stars on the edge are flying around too fast. Gravity, which should be holding the stars in, comes from mass; the more mass an object has, the stronger its gravitational pull. What Zwicky noticed was that the matter he saw in the center of galaxies did not seem to have enough mass to pull in the fast-moving stars on the edges. With the kind of force coming from the center, they should be flying off into space; instead, they keep traveling in a nice circle.

He proposed that there's more matter in galaxies than meets the eye. There's some unseen "dark matter" holding galaxies together. Since Zwicky's conjecture, astronomers have used the dark matter theory to explain more and more unexplained phenomena. Physicists now believe that it makes up the vast majority of the matter in the universe—and that there might be just enough to keep the universe expanding at its current rate, instead of expanding so fast that matter can't stay ac***ulated.

Nevertheless, dark matter remains a profound mystery. Nobody knows what it is, and we've never (we think) observed any in a laboratory on Earth.

Can You be Too Far Outside the Box?

At this point, many students of the history of science may begin to smell a rat. Several times, the vast majority of scientists have believed in some unknown and unobserved substance, only to be proven wrong once some genius comes along. Before Albert Einstein's theory of relativity, almost all scientists believed in a substance called "ether" that filled all of space. The reasoning was that light was a wave, and waves need to travel in something—hence the ether. As soon as Einstein's paper came out, they realized that there was no such thing.

Left: NASA; Right: Weizmann Institute
Moti Milgrom (right), an Israeli astronomer, believes that if Newton's second law of
motion were tweaked just a little, there would be no need for "hypothetical" dark
matter to explain the universe.

An Israeli astronomer named Moti Milgrom applied this intuition to the dark matter question. Maybe it wasn't extra matter that was missing; what was lacking was a proper theory. Perhaps if the laws of physics are tweaked just a little, we'll realize that everything is fine as it is—there's no need for dark matter.

Milgrom proposed that the problem lies in Newton's second law of motion. That law states that the force exerted on an object, F, equals the object's mass, m, times the amount that the object is accelerated, a, succinctly, F = ma. The harder you push on an object, the more its speed will increase; the more massive an object is, the less its speed will increase, or the harder you'll have to push it.

The idea would be that F = ma holds almost all the time—that is, it holds whenever accelerations are sufficiently large, where sufficiently large can in fact be quite small. But for objects whose speed is increasing very slowly, which have a very small acceleration, the law is slightly different. Newton's second law for small accelerations would be something like F = ma².

Of course, the law doesn't immediately switch from one equation to the other. Really, there's one equation that governs everything; it's just that when accelerations are larger, the equation comes out to be very close to F = ma, while when accelerations are very small, the acceleration seems to be squared.

Milgrom did not specify what this function should be—perhaps feeling that it was best to let it be determined by experiment. But to get a better handle on what he is suggesting, it may help to give a specific example, picking a function that exhibits the properties he calls for. This is not too hard to do.

Consider this: let ƒ(a) = (a/a0) ÷ [(a/a0) + 1], where a0 is a constant that is very small. Does this function do more or less what we want it to do? Well, suppose a—the acceleration—is some number that's not too small. If we divide it by a0, which is very small, we will get a large number: if you divide 5 by .001, you get 5,000, for example. So, with a0 = .001 and a = 5, we have ƒ(5) = (5/.001) ÷ [(5/.001) + 1] = 5000 ÷ (5000 + 1) = 5000/5001 = 1 (almost). So, we would get F = ma times ƒ(a) = ma (basically). On the other hand, if a is quite small—smaller than a0—then ƒ(a) will give us a value that is also small, and so close to a. For example, take a (the acceleration) equal to .00001. This gives us ƒ(.00001) = (.00001/.001) ÷ [(.00001/.001) + 1] = .01/(.01 + 1) = .01/1.01 = .01 (almost). Now, 0.01 is bigger than the value for a we started with—.00001—but it's still pretty small, and if we had picked an even smaller a—say, a = 10^(-15)—then ƒ(a) would have been still closer to our starting a. In any case, for very small a, F = ma times ƒ(a) would be reasonably close to F = ma².

Milgrom called his new theory Modified Newtonian Dynamics, or MOND. MOND's predictions ended up fitting a lot of astronomical data even better than the predictions made using the dark matter hypothesis.

Still, most of the astronomy community stuck with dark matter. It takes a lot to change a physicist's mind when it comes to Newton's laws. But recently, mainstream astrophysics scored a big hit against MOND, when an analysis of a collision of clusters of galaxies yielded what looked like direct evidence for the existence of dark matter.

[Essayist]

Well, Let's Take a Look

Now, a group of physicists has tested MOND directly, in a lab. Does F = ma break down when accelerations are very small?

The physicists, led by a group of University of Washington scientists that calls itself the "Eöt-Wash Group," and has made a name for itself by testing basic laws of classical physics at their extremes, used a pendulum to test Newton's second law. The pendulum is not quite like one you might find in a grandfather clock—instead, it is a "torsion pendulum." A thin cord suspends a disk, which lies parallel to the ground. The disk rotates back and forth, clockwise and counterclockwise, like a child on a swing whose ropes have been twisted up with each other.

When the cord is most twisted, and the disk is momentarily at rest, the cord pushes the disk into motion. It exerts a force on the disk, and the disk responds by accelerating, beginning to spin faster and faster. As the disk speeds up, the cord continues to push it, and the disk continues to accelerate.

Once the cord is unwound, it stops pushing. But now, the disk is spinning fast—and not about to stop. The situation is reversed—the disk now begins to push the cord, causing it to twist in the other direction. Meanwhile, the cord is pushing back on the disk, causing the disk to decelerate, or accelerate in the opposite direction, until the rotation stops. Then, the spinning starts all over again. The whole process is dominated by forces and accelerations—a perfect testing ground for F = ma.

Eöt-Wash Group
The Eöt-Wash Group used a pendulum to test Newton's second law. This "torsion
pendulum" has a thin cord that suspends a disk, which lies parallel to the ground. The
disk rotates back and forth, clockwise and counterclockwise.

The amount that the cord twists in the first place determines how much forcing and accelerating is going to take place. The more the cord is twisted, the harder it pushes on the disk, and the more the disk accelerates. By starting the experiment off by only twisting the disk slightly, the physicists were able to keep all of the forces, and hence all of the accelerations, very small.

In addition, the accelerations and the forces are the smallest when the cord is the least twisted. So by focusing on the period when the cord is barely twisted, the physicists were able to study the system when accelerations were very small.

Granted, when a system is spinning, the equations come out slightly differently then when a system is moving in a straight line. F = ma doesn't directly apply—the "force" that spins the disk is actually called a "torque" in physics. But since F = ma is such a fundamental law in classical physics, everything derives from it—including the laws for circular motion. This experiment was similar enough to one involving straight lines that it could help physicists determine whether or not F = ma holds.

And the physicists discovered that it does. As small as the accelerations got, F = ma held perfectly.

MOND believers don't have to give up quite yet. For one thing, it's possible that the accelerations that this experiment tested weren't small enough. But more importantly, as the experiment was taking place, Earth was rotating on its axis and orbiting the sun. So there were other forces and accelerations present, that may have changed the results. MOND's ideas hold only in a reference frame with no extra forces, and no extra accelerations.

Still, these results seem pretty strong. For most physicists, it's hard to believe that Earth's rotation could have made that big a difference. But MOND's successful predictions in astronomy are hard to ignore, and for those who still hold fast to it, this experiment can be useful. Any future version of MOND must agree with the pendulum experiment results.

Though MOND is not quite dead, it's not doing too well either.

In general, which seems the more scientific approach: to change a well-established theory, or to assume the existence of unobserved matter? Can you think of points in history when each approach has worked? Does this experiment change your mind about anything?

Journal Abstracts and Articles

"Laboratory Test of Newton's Second Law for Small Accelerations." link.aps.org/abstract/PRL/v98/e150801.

Bibliography

Cho, Adrian. "No Twisting Out of Newton's Law." ScienceNOW, (April 13, 2007) [accessed May 2, 2007]: http://sciencenow.sciencemag.org/cgi...ull/2007/413/2.

Gundlach, J. H. et al. "Laboratory Test of Newton's Second Law for Small Accelerations." Physical Review Letters, (April 13, 2007) Vol. 98, num. 15, page 150801.