Exploring Mechanics / 8. Oscillations
Nearly anything given a small nudge away from a stable resting point oscillates back and forth in exactly the same mathematical way.
Make a prediction first: a mass on a spring is pulled twice as far from its resting position before being released. Does it complete each full back-and-forth cycle twice as slowly, since it now has twice as far to travel?
x(t) = Acos(ωt + φ) · ω = √(k/m) · T = 2π/ω
The answer to the prediction is no — and that's the single most surprising fact about simple harmonic motion. Amplitude A doesn't appear anywhere in the period formula T = 2π√(m/k). A bigger swing does have farther to travel, but it also picks up proportionally more restoring force and speed along the way, and those two effects exactly cancel.
Newton's Second Law with a linear restoring force gives m(d²x/dt²) = −kx, or d²x/dt² = −(k/m)x. The function that keeps reproducing itself (times a negative constant) under differentiation twice is exactly sine or cosine, which is why x(t) = Acos(ωt + φ) solves the equation for ω = √(k/m) — you can check this by differentiating it twice and substituting back in.
Change the mass or spring stiffness and watch the period change — but change only the amplitude, and the period stays exactly the same, confirming the prediction above. The two bars track kinetic and potential energy trading places, always summing to the same total.
Instead of plotting position against time, plot velocity against position at the same instant. SHM always traces a perfect ellipse here — a single, elegant fingerprint of "restoring force proportional to displacement."
With x = Acos(ωt) and v = −Aωsin(ωt), then (x/A)² + (v/Aω)² = cos²(ωt) + sin²(ωt) = 1 — the equation of an ellipse with semi-axes A and Aω, for any amplitude or frequency at all. The system's entire future is completely captured by where its single dot currently sits on that curve.
Make a prediction first: a heavy bowling ball and a light tennis ball hang from identical strings of the same length. Swung gently, does one swing back and forth faster than the other?
T = 2π√(L/g) (for small swing angles)
Mass doesn't appear anywhere in the small-angle period formula — the answer to the prediction is no, they swing at the same rate, regardless of weight. That's the same mass-cancellation trick from free fall, and for the same underlying reason: a heavier bob feels proportionally more restoring force and proportionally more inertia resisting it, and the two cancel exactly.
Set a length and a planet's gravity, then use the on-screen swing count to check your own reaction-time-based timing against the exact formula.
"Small angle" is doing real work in that formula. Push the release angle up toward 90° and watch the true period pull noticeably ahead of the constant-T approximation.
The exact large-angle period is T = T₀(1 + θ₀²/16 + 11θ₀⁴/3072 + ⋯), with θ₀ in radians — a correction that grows with the square of the angle, which is why it's negligible for a small nudge but very much not for a wide swing. Separately, a real swinging object with size and shape (a "physical pendulum" — a swinging rod, a person on a swing) follows the same form with L replaced by I/(md), where I is its moment of inertia and d is the distance from the pivot to its center of mass — the "simple pendulum" (a point mass on a massless string) is the special case where all the mass already sits at that one distance.
Release the exact (nonlinear) pendulum and the small-angle approximation side by side from a wide starting angle, and watch them fall out of sync — the two only stay locked together for genuinely small swings.
Make a prediction first: a car's shock absorbers wear out and provide less resistance than they should. Does the car now bounce back to level faster after a bump, or slower?
ζ = c / (2√(mk)) — the damping ratio sorts every case into one of three regimes
Counterintuitively, less damping doesn't mean a faster return to rest — it means the car overshoots level and oscillates before settling (underdamped). The fastest possible return without any overshoot at all is exactly ζ = 1 (critically damped) — piling on more damping than that (overdamped) actually slows the return back down again, just without any bounce.
Slide the damping ratio through all three regimes and watch both the motion and the decay envelope change character.
Underdamped (ζ < 1): x(t) = Ae−ζωntcos(ωdt), with ωd = ωn√(1−ζ²) — an oscillation inside a decaying envelope. Critically damped (ζ = 1): x(t) = (A + Bt)e−ωnt — the fastest non-oscillating return. Overdamped (ζ > 1): a sum of two decaying exponentials, both slower than the critical case's — piling on damping trades oscillation for sluggishness.
Same three regimes, now as an actual worn-vs-fresh shock absorber: watch what "less damping" really looks like on the road.
Make a prediction first: you push a playground swing at exactly its own natural back-and-forth rate versus at some random, mismatched rate. Does the timing of your pushes actually matter, as long as you're putting in the same effort?
Steady-state amplitude: A(ω) = F₀/m / √[(ωn²−ω²)² + (2ζωnω)²]
Timing matters enormously — that's the whole answer to the prediction. Push at nearly the natural frequency and even gentle pushes build up a large steady swing; push at a badly mismatched rate and you mostly fight the swing's own motion, getting a disappointing amplitude no matter how hard you push.
Slide the pushing rate away from the natural frequency (1.0× on the slider) and watch the steady swing amplitude collapse — along with the timing between your push and the swing's own motion drifting out of sync.
δ = atan[2ζωnω / (ωn² − ω²)]. Far below resonance (ω ≪ ωn), the motion follows the push almost instantly (δ ≈ 0). Far above resonance (ω ≫ ωn), the motion lags a full half-cycle behind (δ ≈ 180°) — you're effectively pushing against the swing more often than with it, which is exactly why the amplitude is small there too. Exactly at resonance, the lag is precisely 90°, the quarter-cycle timing that turns out to pump in energy most efficiently.
Make a prediction first: soldiers marching in perfect step across a lightweight footbridge have, historically, occasionally caused the bridge to sway alarmingly or even fail — while the same soldiers walking normally (out of step) across it would not. What's different about marching in step?
Resonance: driving frequency ≈ natural frequency ⟹ amplitude peaks
Marching in step means every footfall is a push delivered at the exact same frequency, over and over — the last tab's amplitude formula peaking right where driving frequency matches the bridge's own natural sway frequency. Normal out-of-step walking spreads footfalls across essentially random timing relative to the bridge's motion, which barely pumps in any energy at all.
The same driving-frequency sweep from the last tab, now with multiple damping levels drawn together. Low damping doesn't just raise the peak — it makes the peak dangerously narrow, meaning even a small mismatch between driving and natural frequency still gives a huge response.
Maximizing A(ω) with respect to ω gives the peak exactly at ωpeak = ωn√(1 − 2ζ²), slightly below the natural frequency itself, with peak height growing roughly as 1/(2ζ) for small ζ. As ζ → 0, the peak frequency creeps back toward exactly ωn and the peak height blows up without bound — an undamped system driven exactly at resonance has no steady-state amplitude limit at all.
The Tacoma Narrows Bridge (1940) and the original Millennium Bridge's pedestrian-induced sway are the textbook cases: a structure's natural frequency happened to land inside a range that wind or footsteps could realistically drive it at.
Make a prediction first: two double pendulums are released from angles that differ by less than a tenth of a degree — close enough to look identical to the eye. A minute later, are they still swinging in near-identical paths, or could they be doing something completely different from each other?
Two tuning forks at slightly different pitches, played together, produce a slow "wah-wah-wah" throb called beats — a simple, exactly-solvable stand-in for the same energy-sloshing idea behind two coupled pendulums trading swing amplitude back and forth. A double pendulum, meanwhile, is simple to describe and almost impossible to predict for long: it's chaotic.
Bring the two frequencies close together and watch the combined wave's envelope throb slowly — that slow envelope rate is exactly the difference between the two frequencies, nothing more exotic than simple addition of two waves.
sin(2πf₁t) + sin(2πf₂t) = 2cos(2π (f₁−f₂)⁄₂t) sin(2π (f₁+f₂)⁄₂t). Read the right side as a fast oscillation at the average frequency, whose amplitude is itself slowly modulated by a cosine at half the frequency difference — since amplitude modulation doesn't care about sign, the perceived beat rate is the full difference |f₁ − f₂|.
Two double pendulums, released a hair's breadth apart in starting angle. Watch how long they stay looking alike — and how total, unrecognizable the split becomes afterward. Nothing random is happening in the equations at all; the unpredictability is entirely built from the ordinary nonlinear physics compounding on itself.
The double pendulum's equations of motion are completely deterministic — given exact initial conditions, the future is, in principle, exactly fixed. What makes it chaotic is that nearby starting points separate exponentially over time (roughly Δθ(t) ≈ Δθ₀eλt for a positive "Lyapunov exponent" λ), so any real-world uncertainty in your starting angle — no matter how tiny — eventually grows to swamp any prediction. It's the polar opposite of the orbit simulator from Section 6, where nearby starting speeds gave smoothly, predictably different (not wildly diverging) ellipses.
Two pendulums linked by a spring have exactly two motions that repeat cleanly on their own (the normal modes) — swinging together, or swinging opposite each other. Push just one and watch the energy slosh back and forth between them instead.
Two perpendicular simple harmonic motions, one on each axis — change the frequency ratio and watch the combined path snap between different closed loops, the same patterns behind classic oscilloscope displays.