If you’ve taken a high school physics class, you probably started by learning about position, velocity, and acceleration, ubiquitous concepts in physics that are also well-motivated by our daily life experiences. But soon after that, the course probably moved on to less familiar concepts, such as energy and simple harmonic oscillation modeled by masses on springs. At this point in the class it is natural to wonder: “how is any of this useful? I’ve never even seen a mass on a spring, so why are we spending a whole chapter focused on it?”

And yet, as you continue in your physics education, you’ll see the simple harmonic oscillator over and over again. From models studying the motion of a pendulum to the currents in electrical circuits, all the way down to the theories of the fundamental particles that comprise them, the same equations of motion that describe a mass on a spring show up in all of their grandeur. I’d like to discuss why the simple harmonic oscillator is at the core of so many different models in physics. In order to do so, bear with me as I recall a concept from calculus: the Taylor series.

The Taylor series is a way of writing any well-behaved function as a polynomial – a sum of various powers of x. Calculus teaches us that functions can be approximated by a line if we calculate their derivative and use this as the slope of that line. For values of x near the point where the line is tangent to the curve described by y = f(x), this is a good approximation, but it quickly becomes worse as the two curves diverge. We can improve this approximation, making it diverge less, by instead using a parabola or, even better, a cubic or higher-degree polynomial.

Screen Shot 2023-06-27 at 12.08.40 PM

In physics, this is extremely useful whenever it is difficult to consider a problem exactly – in these cases we use a Taylor series (often using only a linear or quadratic approximation) to solve an approximate version of our problem. Making such an approximation is also well-motivated, because often the function we  wish to approximate is the potential energy of a system, and typically systems do whatever they can to minimize this potential energy (balls roll down hills, springs push/pull masses back towards equilibrium, and radioactive nuclei decay over time). This means that often we only care about a small region around a local minimum of the potential energy function, and such a region can be well-approximated by a parabola.

Screen Shot 2023-06-27 at 12.12.34 PM

Why is it important that a parabola is a good approximation? Because a parabola is the potential energy curve for a simple harmonic oscillator, such as a mass on a spring! Recall the formula for spring potential energy:

Screen Shot 2023-06-27 at 12.13.19 PM

where k is the spring constant describing the stiffness of the spring and x0 is the equilibrium position where the spring exerts no force. 

In summary, because physical systems tend to minimize their energy, the potential energy curves of a huge variety of systems can be well-approximated by the same curve that describes a simple harmonic oscillator. As a result, the behavior of these systems is described by the same equations of motion as a mass on a spring! The relevant variables might change – for example, instead of a mass oscillating back and forth, the equations could describe a string vibrating up and down, or a voltage rapidly switching from positive to negative, or even an electron jumping back and forth between two quantum states – but the underlying math is universal.

This is why high school physics courses teach about mass-spring systems. Although masses and springs themselves might not be very interesting, the simple harmonic motion they undergo provides a foundation for describing phenomena across all of physics. So the next time you’re trying to understand a physical system in terms of a potential energy curve, try to find an analogy with the simple harmonic oscillator. Eventually you’ll start seeing mass-spring systems everywhere!

Ryan is a PhD student at MIT researching material properties and how AI can be used to solve problems in science. Previously, he attended UC Santa Barbara as a physics major where he was part of a small, accelerated program of study designed for highly motivated students.


academics study skills MCAT medical school admissions SAT college admissions expository writing English strategy MD/PhD admissions writing LSAT GMAT physics GRE chemistry biology math graduate admissions academic advice law school admissions ACT interview prep language learning test anxiety career advice premed MBA admissions personal statements homework help AP exams creative writing MD test prep study schedules computer science Common Application mathematics summer activities history philosophy secondary applications organic chemistry economics supplements research grammar 1L PSAT admissions coaching law psychology statistics & probability dental admissions legal studies ESL CARS PhD admissions SSAT covid-19 logic games reading comprehension calculus engineering USMLE mentorship Spanish parents Latin biochemistry case coaching verbal reasoning AMCAS DAT English literature STEM admissions advice excel medical school political science skills French Linguistics MBA coursework Tutoring Approaches academic integrity astrophysics chinese gap year genetics letters of recommendation mechanical engineering Anki DO Social Advocacy algebra art history artificial intelligence business careers cell biology classics data science dental school diversity statement geometry kinematics linear algebra mental health presentations quantitative reasoning study abroad tech industry technical interviews time management work and activities 2L DMD IB exams ISEE MD/PhD programs Sentence Correction adjusting to college algorithms amino acids analysis essay athletics business skills cold emails finance first generation student functions graphing information sessions international students internships logic networking poetry proofs resume revising science social sciences software engineering trigonometry units writer's block 3L AAMC Academic Interest EMT FlexMed Fourier Series Greek Health Professional Shortage Area Italian JD/MBA admissions Lagrange multipliers London MD vs PhD MMI Montessori National Health Service Corps Pythagorean Theorem Python Shakespeare Step 2 TMDSAS Taylor Series Truss Analysis Zoom acids and bases active learning architecture argumentative writing art art and design schools art portfolios bacteriology bibliographies biomedicine brain teaser campus visits cantonese capacitors capital markets central limit theorem centrifugal force chemical engineering chess chromatography class participation climate change clinical experience community service constitutional law consulting cover letters curriculum dementia demonstrated interest dimensional analysis distance learning econometrics electric engineering electricity and magnetism escape velocity evolution executive function fellowships freewriting genomics harmonics health policy history of medicine history of science hybrid vehicles hydrophobic effect ideal gas law immunology induction infinite institutional actions integrated reasoning intermolecular forces intern investing investment banking lab reports letter of continued interest linear maps mandarin chinese matrices mba medical physics meiosis microeconomics mitosis mnemonics music music theory nervous system