How not to do your physics homework


Foucault’s pendulum has fallen. On April 6, the steel cable snapped and sent it crashing onto the polished floor of the Musée des Artes et Metiers in Paris. The 28 kilogram brass weight ended its 159-year career—the dented bob is, a museum spokesperson affirmed, beyond repair—doing what it was meant to do: obeying the law of gravity. I have to admit I shed a tear (or at least the idea of a tear) for the fallen bit of scientific history, not because I’d visited the pendulum myself, or even read the 1988 Umberto Eco novel which takes its title and climax from the now-not-swinging orb. I have my own tangled history with pendulums—one stretching back, depending how you count it, decades, even centuries. It’s quite a bit of weight to bear, but a tale worth telling.

Fifteen years ago, I took a physics course in “classical mechanics”—the sort of stuff figured out by Newton and his contemporaries: laws of falling bodies, what happens when billiard balls bounce off one another, that kind of thing. The whole course was mostly word problems and formulas—math aided by the occasional diagram. At the end of the semester, though, there was one lab exercise: we were supposed to construct a pendulum, and then use it to measure g, the earth’s gravitational constant—the acceleration due to gravity, or how fast an object gains speed as it falls. If you know the length of a pendulum, and how long it takes to swing from side to side (think of the ticking of a grandfather clock), you can calculate g. Of course there’s no real need to calculate g—Galileo did this for us, three hundred years ago, experimenting with dropped balls of different weights. What he found out was that falling bodies, regardless of their weight, accelerate at the same rate—which is, when you switch it into the metric system, 9.8 meters per second, per second. What Galileo discovered by dropping balls, we physics students were to verify and document with our swinging weights.

I should say from the start that when our professor assigned us this project, he was not out to make things particularly difficult or complicated. The idea was that we would string our pendulums in dorm rooms or campus stairwells—record the data, make the calculations, write them up, and be done in an hour or two. Gravity is a simple and consistent force, but for my friend Dave and me, the pendulum would swing another way, driven by forces less predictable and—for us at least—more powerful than those of simple science. The laws of physics are the same everywhere. But for the purposes of this story, place becomes not just relevant, but essential.


The Death of General Warren at the Battle of Bunker Hill, by John Trumbull

Two hundred thirty-five years ago, during the first year of the American Revolution, Boston was under siege. The British army, headquartered in the city, was more powerful than the colonial fighters but remained cut off from the surrounding countryside. In June of 1775, the loyalist troops moved to expand the area under their control, taking a strategic hill or two on the perimeter of the city. When the colonial fighters heard that the British were aiming to the high ground in Charlestown, across the river from the city center, they erected some preemptory fortifications on Breed’s and Bunker Hills and laid in wait. The battle took place on June 17: the rebels fought fiercely, but eventually the British broke through their lines and overran Breed’s Hill, their highest redoubt, battling the retreating rebels across the lower crest of Bunker Hill as they retreated to Cambridge to regroup. The British got their ground, but at great cost, winning the day but sustaining the more significant casualties. They even lost the battle’s name, which went not to lofty Breed’s but to Bunker, where the colonials had done less fighting but sustained the larger portion of their losses.

A few weeks before our project was due, Dave had walked the Freedom Trail, a collection of historical sights around downtown Boston. He’d paused at the site of the Boston Massacre, taken in the Old North Church (which sent the signal that started Paul Revere on his famous ride), and, in a little park overlooking Charlestown, climbed the Bunker Hill Monument: an obelisk of Quincy granite, the same shape and half the height of the Washington Monument. When it was completed in 1842, it was the tallest structure in North America.

I don’t know exactly David was thinking as he ascended those steps—about the battle and its historical importance (a Pyrrhic victory for the British, a surprise confidence-booster for the colonials), perhaps about architecture, civic pride, the nature of monuments themselves. But when he got to the top, and peered down the central well descending straight to the tower’s base, he saw not history, but scientific inspiration.

The longer a pendulum is, the slower it swings. The slower it swings, the more accurate the measurements you can make of its cycles, and the more refined a calculation can be made for g: g = 4π2L/T2

Our assignment was to construct a pendulum a few feet long; what Dave saw was the chance to swing one a hundred times longer, turning the very forces of American history and memory towards the task of scientific measurement. When he pitched the idea to me a few days later, I immediately recognized its brilliance. It was overkill, but science rewards ambition—as would, we assumed, our awe-struck professors and fellow students. So Dave put in a call to the National Park Service. Somewhat surprisingly, our request was granted.

In 1825, as the 50th anniversary of the Battle of Bunker Hill approached, the good citizens of Boston and Charlestown decided that something needed to be done to commemorate the revolutionaries’ victorious defeat. Committees were formed, pledges were made, a $100 design competition was held. The winner was Solomon Willard, a Boston stone-carver and model-maker who had submitted a classical obelisk of granite bricks, nearly double the height of the tallest ancient Egyptian model. A grand ceremony was held on the June 17 anniversary, the Marquis de Lafayette (in town for the weekend) laid the cornerstone, and then-congressman Daniel Webster gave one of his trademark orations:


We wish that this column, rising toward heaven among the pointed spires of so many temples dedicated to God, may contribute also to produce, in all minds, a pious feeling of dependence and gratitude. We wish, finally, that the last object to the sight of him who leaves his native shore, and the first to gladden him who revisits it, may be something which shall remind him of the liberty and the glory of his country. Let it rise! let it rise, till it meet the sun in his coming; let the earliest light of the morning gild it, and parting day linger and play on its summit.

Of the day itself, here’s what I remember: it was one of those hazy midwinter Boston days when there’s snow on the ground but only in dirty patches, and everyone pretty much stays inside. We got ourselves to Charlestown by bus and then by foot, ascended the hill and stepped inside the stately classical visitor center near the obelisk. The ranger on duty welcomed us and, after a brief chat, escorted us to the monument. We walked through the studded wooden doorway and into the tower’s base, and stood in the cold stone alcove at the base of the stairs. As the ranger unlocked the iron gate that keeps people out of the tower’s central well, he told us that a professor in the 1850s set up a pendulum in the tower to prove something or other about the manner of the earth’s rotation. We took this as a sign of the rightness and nobility of this venue. He also told us that by sometime in February, the door to the monument would freeze shut, and they’d have to wait until it thaws before visitors could go up again. With that, he wished us luck and retreated back into the heated visitor center.

We paused for a minute to survey the situation and breathe in the icy monumentality of the task at hand. Craning our necks, we could see a tiny slatted circle centered above us; all else was blackened stone. The chamber, seven feet in diameter, felt a little crowded with the two of us plus a large pillar-and-plaque replica of the first monument erected on the battle site—the replica, like the original, commissioned by King Solomon’s Lodge, a local Masonic group. We took out our supplies: a spool of 12-pound-test nylon fishing line, a three-liter soda bottle which Dave had rescued from his dorm room trash can and I’d patiently refilled at the visitor center’s water fountain; our lab notebooks; my stopwatch; a couple of ballpoint pens. I pocketed the line and we began the long ascent.


Two years after Lafayette laid the cornerstone they had to dig it up and lay it again, a task made easier by the lack of anything yet built above it. The 1820s gave way to the 1830s, the 1830s to the 1840s, and the monument grew only haltingly, a course or two of granite at a time (huge blocks brought down from Quincy on a horse-drawn railroad) and then for years on end, no progress. Amidst the pointed spires of Boston, the incomplete Monument was just a stub on a hill, fast becoming a national embarrassment. Property owners on the lower slopes talked of buying out the site and tearing the thing down to save the neighbourhood. Luckily for Charlestown (and for the history of American feminism), the women of Boston decided enough was enough: they formed their own committees, sent out appeals, and achieved what their husbands could not. Funds were raised, the work recommenced, and the capstone was laid during the summer of 1842, and on June 17, 1843, on the 68th anniversary of the battle (and the 18th of the groundbreaking), the Bunker Hill Monument was dedicated. Lafayette was unavailable (he’d died in 1834), but Daniel Webster was brought back to deliver another lengthy oration. After a brief rundown of the monument’s builders and funders (including a much-deserved shout-out to “the winning power of the sex” who’d brought in the final needed cash), Webster dashed the hopes of those waiting for him to top his first Bunker Hill address, suggesting that the greater eloquence lay not with himself but with the stack of granite behind him:

It has a purpose, and that purpose gives it its character. That purpose enrobes it with dignity and moral grandeur. That well-known purpose it is which causes us to look up to it with a feeling of awe. It is itself the orator of this occasion. It is not from my lips, it could not be from any human lips, that that strain of eloquence is this day to flow most competent to move and excite the vast multitudes around me. The powerful speaker stands motionless before us.

We climbed the 294 spiral steps to the top of the tower, a granite chamber lit by four rectangular windows affording a dingy but still beautiful panorama—Charleston, Chelsea, Somerville, Cambridge, Boston; the Charles and Mystic Rivers; the harbour leading out to Massachusetts Bay. The ceiling overhead was a smooth stone vault, pointed at the center like a gothic arch. At the apex a thick iron staple was fixed into the stone. Here was where the former pendulum must have hung; sadly for us, it was too high to reach. We turned our attention to the iron grate in the middle of the floor, set in the pavement like a four-foot manhole cover. By kneeling down and squinting through the bars, you could just make out the dim light coming in from the bottom of the well.

The fishing line spool wouldn’t fit through the grate, so we tied my keychain to the end of the line and, as Dave began to lower it, I rushed back down the stairs to meet it at the bottom. I gave a shout to Dave, who tied off the top of the line to the metal grate and tromped down the spiral steps to join me.

Our first attempts to hang our bottle-bob were unsuccessful; the line kept stretching each time we retied, lowering the pendulum to the floor. Finally, though, we got it steadily hovering waist-high—a scientific instrument of incredible accuracy, constructed of plastic and liquid, and set to swing at the center of one of the United States’ national treasures. If it hadn’t been so bitterly cold at the center of that dark stone well, I would have wept at the beauty and synchronicity of it all.

In the winter of 1851, in Paris, the forty-two year old physicist Jean Bernard Léon Foucault hung a six-and-a-half foot pendulum in his cellar laboratory, and with careful measurements of the movement of its swinging arc offered experimental proof of the earth’s rotation. Over the next few months he repeated the experiment with ever longer pendulums in ever more exalted venues, till on March 31, at the Emperor Louis Napoleon’s request, he swung with great acclaim his twenty-eight-kilogram brass sphere from a 220-foot wire strung from the domed ceiling of the Paris Pantheon. “Thence,” reported Putnam’s Magazine a few years later, “throughout the world, a pendulum mania extended, until a monster pendulum threatened to become essential in every respectable household.”


Eben Norton Horsford, recently having been appointed to a professorship in the applied sciences at Harvard, was a respectable man, and when the news of Foucault’s triumph reached American shores, he knew what he had to do. With the university’s backing, the permission of the Monument’s caretakers, and the help of the Massachusetts Charitable Mechanics Association, Professor Horsford had his own monster pendulum swinging in the Bunker Hill Monument by mid-summer. His preparations were somewhat more exacting than Dave’s and mine: he had the grating lifted from the top of the central well and the monument-within-a-monument removed from the bottom chamber, allowing the pendulum to hang unimpeded from the capstone staple. All the ventilation holes between the well and staircase were sealed, and wooden structures were added top and bottom to stop any air currents from interfering with the pendulum’s swing. The Charitable Mechanics constructed an elaborate sighting apparatus at the base to allow for precise observation of the pendulum’s motion. It was only in selecting the weight that Professor Horsford allowed location to get the better of him, choosing a 32-pound cannonball that had been fired by the British that June day in 1775. This he set into motion using Foucault’s own exacting method, pulling it back with a single thread, which was then burned with a candle to release the weight, minimizing the chance of human interference in its motion.

I gingerly pulled the soda bottle a couple feet to the right and simply let it go. Very, very slowly the pendulum began to swing—or rather to move eerily back and forth across the chamber. We started to realize that maybe things wouldn’t be as simple as we’d thought. The whole advantage of having such a long pendulum was that it would let us make incredibly accurate measurements of its rate of oscillation. The problem with this was, though, that it moved so slowly that it was hard for us to figure out just when our weight changed direction at all. At the peak of each swing it just seemed to hang there, motionless, for several seconds, before slowly starting to inch back towards the center of the well. And, as I said, it was dark, and it was cold. My fingers were so numb I could barely work the stopwatch.

We discovered another problem: 195 feet of fishing line has a lot of bounce in it; our pendulum moved up and down as well as side to side. This meant, as any student of classical physics will tell you, that our pendulum was also a spring, subject to its own harmonic equation: T=2π√(m/k)

We had no idea how to combine the two formulae, so Dave and I gathered our waning excitement and tried to make the best of things: steadying the bottle to minimize the vertical motion, and getting what we thought were one or two good stopwatch measurements of its oscillation before deciding we were done. Dave sprinted up the stairs to cut the pendulum loose; halfway up, I heard his echoing footsteps slow and cease, replaced by pained gasps.

“Are you ok?” I shouted.

“Yes . . . just . . . ran . . . too . . . fast . . . ” came the wheezing reply. Visions of heart attacks danced in my head; I pictured myself at the memorial service, delivering the eulogy: “He died young, but at least he died for science.” Dave made it to the top, but our mutual lightheadedness probably clouded our judgment in what we did next. I don’t know what we’d expected that much fishing line to do when we let it drop down the well—coil itself neatly?—but when Dave cut the line, what landed at my feet a second or two later was a foot-wide hairball of inextricably tangled monofilament. It was late, and the gravity of our error did not sink in. We packed up our stuff, thanked the ranger, and headed back to campus as the sky grew dark.

Professor Horsford had his troubles too. The cannonball didn’t swing straight; its behaviour was at times erratic. The sighting device only worked from certain angles. The convenient staple in the tower ceiling restricted the pendulum’s motion. He worried about the time it took him to look from the pendulum to his pocket watch and back. But it was not the dead of winter; the door was in no danger of freezing shut, and so he persevered, tested and retested, figured out ways around his difficulties, tried additional approaches.

That night, back in the dorm, we regrouped to plug our data into the pendulum equation and revel in our exactitude. But in a warm room, the data did not look so complete. We quickly realized we were missing the very thing that we should have known from the outset: we had no real idea of the most important starting condition, our pendulum’s length. Our initial plan to measure the cut-down line a tape-measure’s-length at a time was foiled by the Gordian knot of nylon in my backpack.

We furrowed our brows. We paced. We bemoaned our fate. But then I had an Alexanderine thought: what if we could measure the line without untangling it? We ran to the organic chemistry lab and commandeered a highly accurate digital scale. By comparing the weight of the ball to the weight of a known length of line, perhaps . . . but every time we ran the numbers we kept getting lengths much taller than the full monument’s height.

Technology failed us, so we turned to history and sprinted for the library. After a few frenzied searches for blueprints in various stacks and archives, the best I managed to find was a cross-section elevation drawing of the structure, showing the well, the staircase and the upper room, but lacking any scale or measurements. I pulled out my National Park Service brochure, which had the official height, and tried to gauge the interior proportions from the drawing. I came up with something between 190 and 200 feet. The whole point of a long pendulum had been accuracy, but now we could only guess the length within five percent.

Still, at least we had some numbers. With tempered pride cranked them through the equation the equation to reveal . . . a value for g in excess of 11 m/s/s, far off our target 9.8. This should have been the end of our endeavour. Alas, rather than letting reality call our data into question, we decided it had to be the other way round. “I know!” David said. “Didn’t they tell us that the value of g varies depending on location? If there’s greater-than-average mass near the point where we’re measuring, that might explain our result! There must be some sort of heavy matter concentrated under the obelisk!” It was a thin thread to hold on to, but hold on we did, all the way to the geology library, where I confidently asked to see any and all magnetometric maps of the Boston area. There was a gigantic, gravity-bending iron deposit under Bunker Hill, and I was going to find it.

Eventually Professor Horsford found the numbers he needed, uncabled his cannonball, and wrote up with a full description of his efforts for the August 1851 meeting of the American Academy of Scientists. There was a longer paper on the Foucault replication, and a shorter one about something he’d noticed along the way, how on sunny days the dead-center point of the pendulum moved a half an inch to the northwest, returning to dead center when clouds or night appeared—evidence of the sun-kissed Quincy stones’ expansion in the heat. It was this discovery, and not the pendulum swing, that Putnam’s Magazine thought worth noting in its report: “Daily, the summit of that proud pile was found to trace an approximate elliptic orbit of about half an inch major axis, offering thus on homage to the great luminary . . . “

Magnetometric maps, it turns out, are not so easy to read, and my iron lode stayed hidden. I tried another library or two for books on gravity, on old New England, on anything tangentially tied to Bunker Hill. By this point, Dave and I had finally given up, retreating to our separate rooms to slog through our separate, deeply flawed write-ups. I filled my lab book’s gridded page with a brief description of our experiment and of the possible reasons for its surprising result. I tried to maintain a scientific distance in the write-up, but I felt my anger coming to a head. I’d completed the assignment in the most impressive way imaginable, but I had the wrong result! I looked around my dorm room for inspiration. Two minutes and thirty seconds later, I was dangling a bar of hotel soap from ten inches of mint-flavoured dental floss. I disdainfully flicked it with my index finger, glared at my stopwatch while the thing went through ten quick oscillations in ten seconds. Ten seconds more and I’d run the new numbers through the equation, arriving at a perfect, accurate 9.8. Now well nigh furious, I rushed my write-up to conclusion with a pithy statement about the irony of big projects gone wrong, put down the pen and shut the book.

During the last review section before the final exam, my graduate student teaching fellow, a kindly former professional ballerina with an alarming habit of tripping over things, returned my opus. Regarding the historical nature of our project, she had no comment. The equations themselves, and the lack of a proper estimation of my percentage error, were all that mattered. I’d missed the point entirely, and was lucky to slip through with a B-minus.

“Suppose,” the Gospel says, “one of you wants to build a tower. Will he not first sit down and estimate the cost to see if he has enough money to complete it? For if he lays the foundation and is not able to finish it, everyone who sees it will ridicule him, saying, eThis fellow began to build and was not able to finish.'” Darn good advice.

A little over a year after Professor Horsford’s pendulum came down, Daniel Webster went to his reward in Marshfield, Massachusetts, thirty miles southeast of Bunker Hill, after uneloquently falling off his horse. His dying words were “I still live.”

Léon Foucault made important discoveries regarding magnetism and optics, was made an officer of the Legion d’Honneur and a member of the Royal Society. He died in 1868.

The Bunker Hill Monument’s national fame soon faded; the excitement surrounding its completion gave way to mild embarrassment. Herman Melville’s 1855 satirical novel Israel Potter: His Fifty Years in Exile included a three-page dedication “TO HIS HIGHNESS THE BUNKER-HILL MONUMENT,” chipping away at Webster’s solemn edifice by engaging the obelisk conversationally (the dedication is signed “Your Highness’s / Most devoted and obsequious, / THE EDITOR”).

It hardly mattered. By then the nation’s obelisk obsession had shifted to the nation’s capital, where construction of the Washington Monument had not only gotten under way but had, in another echo of Bunker Hill, run out of funds and ground to a halt. For twenty-five years, the first president would be memorialized by a crane-topped marble stump at the center of the National Mall. The monument was finally completed in 1888, forty years after groundbreaking.

Ephemera from Horsford’s baking powder empire

Professor Eben Horsford went on to greater, stranger things. A few years post-pendulum, he invented and commenced manufacturing a wildly successful double-acting baking powder. He died in 1893, a wealthy man, having spent his latter years and part of his fortune trying to prove that Boston had been colonized by Vikings 500 years before Columbus. He dug around a bit in his neighbourhood, found some suspicious foundations, and had a granite plaque installed along a Cambridge sidewalk, reading “On this spot in the year 1000 Leif Erikson built his house in Vineland.” A few miles further up the Charles he descried signs of a more massive Norse settlement. To aid the imagination, in 1889 Horsford had a tower built of round grey river-stone, with a central winding stair of 67 steps (no room, alas, for a pendulum), to commemorate the supposed Viking fortress.

Despite our errors in judgment, ambition, and lab practice, Dave and I both did all right in our physics course. After the final exam, Dave stopped by to tell our professor, a rather intimidating Nobel laureate, about our experimental misadventures. The professor laughed and laughed, then listed a dozen possible sources of error that we hadn’t even begun to imagine, and laughed some more.

In the end, though, the hard sciences weren’t for either of us. Dave turned his sights toward American literature, and after a bit more intellectual zigging and zagging, I wound up studying the history of science, having decided that the most interesting part of science was the stories of the scientists themselves. The rest of my years in Boston, whenever I rode the subway as it rises over the Viking-prowed Longfellow bridge, I liked to look out on the Charleston side, taking in the Monument’s still-impressive height, and imagining, beneath its winking red beacon, a long transparent nylon thread, connecting past and present, folly and grace, our most impressive plans and their undoings, tracing movements far beyond the work of mere equations. +

Originally published as “The Stories of Scientists” in Comment Magazine, 1 September 2010

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