Violet Piper

What the Simple Harmonic Oscillator Taught Me

Glimmers

Physics, at its best, is a thrilling, graceful discipline linking the abstract world of mathematics to the concrete universe we live in. At its worst, it's a droning lecture in a massive auditorium conducted by an old man with no intention of engaging the room or even waking them up from their naps. Learning is beautiful; being a student is not.

And in particular, being a physics student is not. I imagine there's at least some glamour in studying something like history: Classrooms hot with debate and a professor's scribblings in the margins of your paper about your flow of ideas, tone, and voice. Physics students don't have voices; we have quivering, outstretched hands and answers to problem sets we each copied off the person to our left.

As an undergrad, I spent most of my time thinking I was the stupidest person in every building I walked into, crying about deadlines and memorizing integrals for exams. There were only a few moments when I looked up at the blackboard or down at my homework and realized how amazing it was to discover and prove How Things Work. Thankfully, those fleeting glimmers of awe were enough to carry me through to the exciting stuff much later.

But, before that, boredom.

—

One's first introduction to physics is littered with manufactured, unimaginative vignettes of blocks on inclined planes, pulleys, levers, billiard balls, and boxes full of gas. These physical systems involve only a few knowns and unknowns, offering a chance to apply the most basic universal principles: Newton's laws of motion, Conservation of Energy, thermodynamic equilibrium, and the like.

"Here is this object," professors say. "These are its most important qualities. I will place it here. Where does it go? How does it change?"

Physics can be boiled down to studying how things change in time—how they evolve. It is the quintessential application of calculus, the math of Rates of Change, and the most metaphorical subject to study during my transition into adulthood.

It's the Weirdest Thing

In college, I began, and since have never stopped, using physics to describe my life. I am often an atom, a neutron star, or a wave in my stories. I tell my therapist that being anxious is my ground state—that I can't access a lower energy level. I imagine arguments with my boyfriend as a vector space—one equivalently described by two different bases, his version of the story and mine. Making plans feels exponential to me and canceling feels linear. Eating feels parabolic, and everything involves some symmetry, constant, or boundary condition.

And somehow, no physics problem has been more visceral, applicable, or informative than the Simple Harmonic Oscillator (SHO). It's the weirdest thing. When things feel complicated, I come back to it; it helps me reposition, simplify, and define.

—

The SHO is another aforementioned entry-level system—the physics equivalent of The Trolly Problem in an Ethics 101 course. The Simple Harmonic Oscillator is introduced one day, sketched onto a whiteboard, and then never really goes away, popping up somewhere in every branch of the subject.

It is a weight resting on a table, with one end attached to a wall via a spring. That's it. In the Real World, which physicists discuss with such disdain, you could pull on the weight, stretching the spring, release, and the weight would bobble back and forth across the table. The weight's range of motion would shrink until it eventually came to a stop.

In Intro to Physics' ideal, drag-less world, where energy doesn't pathetically flutter away through friction, the weight would slide back and forth forever, oscillating perfectly, simply, and harmonically: The Simple Harmonic Oscillator.

I love so many things about this seemingly stale apparatus.

Firstly, it is secretly one-dimensional. Yes, the weight, spring, and table are 3D, but the weight moves towards and away from the wall in a single, straight line. Only one dimension matters—the other two are "degrees of freedom" you can ignore with impunity.

This is what thrills me: The practice of editing, streamlining, and scraping away the noise. You can always choose a different reference frame—a different perspective. Often, there is one that makes life the easiest.

In fact, another preliminary and vital step in physics problem solving is deciding how to label your system—how to keep track of all the moving parts. After all, what is perspective if not a set of labels? Which direction is positive or negative? Where is Zero? Simply putting Zero in the right spot can drastically reduce the time and effort it takes to solve a problem.

Relabeling

College life is a non-stop relabeling. Your roommate is your Best Friend until you find your actual best friend. Dining hall pizza is Inedible until you're drunk. You're a Freshman, and then a Sophomore—an Outsider, and then a Local. Every time you turn to look at your path up until that moment, everything gets a new tag.

"I didn't know anything then, but I know things now." We mark Zero behind ourselves, over and over again.

—

Now, this time, when picturing the SHO, imagine the weight sliding along a number line. If you place Zero where you let go, the weight slides up past some numbers, reaches a maximum value, slides back past those same numbers, and returns to Zero. Similarly, you could place Zero at the other end when the spring is fully compressed.

Both sets of labels are apt, but there is a better one that illuminates the crucial symmetries of the system: Marking Zero, where the weight rests when the spring is not compressed nor extended but at its comfortable, natural length.

However far you pull the weight along the number line before you release it is precisely how far it will travel on the other side, like Newton's cradle. If you release the weight at Four, it will slide to Negative Four and back again. Symmetry!

The system forever ticks between two points of equal, greatest displacement from Zero. It passes through the center between each tick, where it's "minimally displaced" from Zero. Maximum, minimum, maximum. Physics, life, the ubiquitous sloshing dance!

But there is another hidden dance in the weight's velocity. The weight is released from rest—its minimum speed. It gets faster and faster until reaching its max speed in the exact center before slowing again to momentarily stop on the other side.

Neutral

When the SHO and I were first introduced, I was horrified by the notion that for something to change direction, it had to, for a split second, come to a complete stop. I tried to slow the SHO's motion in my mind, but I could not picture it for the life of me. Months later, while a friend was giving me a driving lesson in the campus parking lot, it hit me: The gear shift has to pass through neutral before reversing.

Nothing offers me more peace than the knowledge that, whether I like it or not, I must pass through neutral before changing direction in my career, relationships, or goals. Moments of rest are built-in—they're integral to the system.

—

So, within the SHO, we see a trade-off of position for velocity. Further, slower, closer, faster. If you were to graph these two values through time, you would get these beautiful inverse waves harmonically bending around each other.

As it turns out, velocity and position are "conjugate variables" or "Fourier transforms" of one another and together make up a beautiful "uncertainty relation." In quantum mechanics, knowing where a particle is means to be utterly clueless about how fast it's going. It is a horrifying fact and a foundational rule of our universe—it's the balance.

And still, there is a final dance—the energy dance! The SHO's sloshing motion and sloshing of minimums and maximums are Conservation of Energy in action. Before you release the weight, all of the system's energy is stored in the stretched spring. When the weight passes through the center point, its energy has fully transformed into the kinetic kind—the spring is momentarily relaxed. On the other side, the energy transitions back into potential, now stored in the compressed spring.

Energy

Undergraduates barter their energy like frenzied street vendors. Three hours of sleep and an eight-ounce caffeinated beverage will get me through the start-of-lecture quiz, I would think to myself, sprawled in the shade of a massive sequoia. A brief session of barreling laughter with my peers outside the library could often charge me enough for an ominous meeting with my thesis advisor. My senior year housemate left each morning furiously shaking a protein drink in preparation to hit the gym before his discussion section. Everything was either buzzing or perfectly still for four years straight.

—

In quantum field theories, we treat particles as fields—infinite sets of harmonic oscillators. In classical mechanics, we sometimes model the atoms in solids as tiny harmonic oscillators—little weights connected by springs on all sides, pushing and pulling on each other. As I said, they don't go away. The SHO is like a friend continually returning from study-abroad trips with a new haircut and accent but the same unwavering hold on their values and opinions.

The Endeavor

I continue to move Zero around my life and work hard to strike a balance. I Facetime my friends from college, all in different cluttered bedrooms around the country, and we remind each other about rules we used to follow or that are necessary now. We love to know How Things Work and never entirely do. It's a simple endeavor, a harmonic one.

Violet Piper is a writer, artist's assistant, camp director, and astrophysicist from Brooklyn. She has published essays, poems, and stories in Slate, The Blue Mountain Review, Harpur Palate, and others.