The Harmonic Oscillator — Does a Bigger Swing Take Longer?

Source

Simulation: The Harmonic Oscillator — Why Period Ignores Amplitude (misconception-first, counterfactual two-well mode).
Concept: A particle in V = ½kx² oscillates as x(t) = A·cos(ωt + φ) with ω = √(k/m); the period T = 2π√(m/k) is independent of amplitude, while KE and PE trade off at constant total E = ½mv² + ½kx² = ½kA².
Serves: Reading potential-energy diagrams; SHM as the linear-restoring-force special case; energy conservation and turning points where V(x) = E.
Inquiry: Two balls released together from 0.30 m and 0.60 m — which returns first? Which slider CAN change the period? In which well do the recorder traces dephase?
Misconception target: "Larger amplitude ⇒ longer period." True in a constant-force well V = F₀|x| (T ∝ √A) — false for the spring, where F = −kx grows with the swing and amplitude cancels exactly.
Verified values: k = 20 N/m, m = 0.5 kg → ω = 6.3246 rad/s, T = 0.9935 s at BOTH A = 0.3 m and A = 0.6 m; E(A = 0.3 m) = ½kA² = 0.90 J, all potential at x = ±0.30 m.

Pedagogy principles applied

ID	Principle	How it shows up here
P1	Prediction first, optional	Starts paused at the release instant; T₁/T₂ readouts show "—" until first ▶ Play; Predict panel sits senior to all readouts and never blocks play.
H1/H2	Misconception structural	"Smaller swing home sooner" is a prediction choice, not a standing card; its rebuttal appears only when selected.
CF	Counterfactual contrast	Both wells implemented exactly: ✓ spring T = 2π√(m/k) vs ⚠ constant force T = 4√(2Am/F₀) ∝ √A, F₀ tuned to match at A = 0.30 m.
C1/C2	Readout discipline, SI units	Only T₁ and T₂ visible by default; KE/PE/E numbers live in a collapsed live ledger; SI everywhere, real-time animation labeled.
G4	Responsive controls, stable scales	Fixed axes (±0.78 m, 0–10.5 J, 6 s window) so slider effects are visible; every slider previews live even while paused.
E1	Animation honesty	Closed-form motion (shared phase in the spring well; exact parabolic arcs in the ⚠ well) — no integrator, no period drift.
INV	Invariant vs changing	KE⇄PE capsules keep fixed length (E conserved); E-lines and turning points move with A while T does not.
A3/A4	Formal layer off by default	Equations live in the ⚛︎ The Physics revision sheet, opened on demand from the top bar.
AUD	Audit hook	window.__audit.at() returns T = 2π√(m/k) from the sim's own period function at current sliders.

The Harmonic Oscillator

Teach me Hide Text

Speed

▶ Press Play — ball releases from rest

Guided Inquiry

1 · Two balls, one spring well

Let’s meet our two balls. Both sit at rest in V = ½kx² (k = 20 N/m, m = 0.50 kg). Ball 1 waits at A₁ = 0.30 m, and ball 2 waits twice as far out, at A₂ = 0.60 m. Same well, same mass — only the release points differ.

2 · Predict — the two-ball race

Here’s the race. Released together from rest, each ball swings in, out the other side, and back again. Which one do you think completes a full cycle first?

3 · Watch the periods lock

Let’s see how your prediction holds up. Press ▶ Play if needed. The recorder traces cross zero together, and T₁ and T₂ both read ≈ 0.993 s. Try dragging A₁ and A₂ apart — the lock holds. Amplitude sets the energy and turning points, not the clock.

4 · Predict — what moves T?

So you’ve just seen A₁ ≠ A₂ leave T unchanged. In this spring well, which change do you think does move the period?

5 · Predict — four times the mass

Let’s test that with the mass. Both balls are replaced by ones four times heavier (m: 0.50 → 2.00 kg), in the same spring, from the same two release points. What do you expect the two periods to do?

6 · Amplitude sets energy, not the clock

Let’s put it all together. In the ✓ Spring well, F = −kx grows with the swing, so A cancels from T = 2π√(m/k): only the spring constant k and the mass m move the clock. Amplitude still matters — it sets the energy E = ½kA² and the turning points x = ±A. And that’s the big idea: a period that ignores amplitude is the fingerprint of simple harmonic motion.

Listen

Controls

▾

Spring constant k20 N/m

Mass m0.50 kg

Compare two amplitudes

Amplitude A0.30 m

Amplitude A₂ (ball 2)0.60 m

T = — s

Show x(t) chart

⚡ Energy ledger (live)

▾

KE = ½mv²

PE = ½kx²

E total

● 1

—

Exact solution used to move the balls in the ✓ spring well (φ = 0: released from rest at x = A). The ⚠ well uses its exact piecewise-parabolic solution — no numerical integration anywhere.

The period contains m and k only — amplitude is absent. This is the defining fingerprint of simple harmonic motion.

KE ⇄ PE trade at constant total (values for the current sliders and well). Turning points where V(x) = E ⇒ x = ±A.

⚠ counterfactual well: with the force capped at F₀, the period grows with amplitude. F₀ is tuned so a 0.30 m release matches the spring's period at the current k and m.

The Physics

The harmonic oscillator · what this sim shows, the physics behind it, and every control

At a glance

Two balls in the same spring well are released from rest at different distances from the centre. The sim records both motions, so you can test what sets the time of one oscillation, the quantity a clock counts.

What you'll learn

How the period depends on the release distance, the amplitude A.
How the spring constant k and the mass m enter the period.
How the energy and the top speed depend on A.
Which force law gives simple harmonic motion, and what makes it special.

The same spring well and the same mass for both balls. Only the release distance differs.

What's on screen

Violet curve: the spring's potential energy V = ½kx² against position x.
Ball 1 (cyan) moves along the x axis. Its dot on the curve marks its potential energy; the arrow is the spring's pull on it.
Ball 2 (amber): the same, released from its own distance.
Dashed lines: each ball's total energy E. The ball turns round where its line meets the curve.
x(t) chart: each ball's position over the last 6 s, with faint lines at ±A.
Readouts: the periods T₁ and T₂. The Energy ledger lists each ball's KE, PE and E.

See itRelease both balls from rest, ball 2 twice as far out as ball 1. Watch for: the two traces on the x(t) chart, one twice as tall as the other.

The controls

Press ⌖ on a row to see where that control sits in the sim. Moving any slider releases both balls from rest again.

Control	What it changes	Range
Spring constant k	How stiff the spring is: the pull is k newtons for every metre from the centre.	5–40 N/m opens at 20
Mass m	The mass of each ball. Both balls always share it.	0.2–2 kg opens at 0.50
Compare two amplitudes	Shows ball 2 with its own release distance. Off, only ball 1 swings.	on / off opens on
Amplitude A₁	How far from the centre ball 1 is released.	0.15–0.70 m opens at 0.30
Amplitude A₂	How far from the centre ball 2 is released.	0.15–0.70 m opens at 0.60
Show x(t) chart	Shows or hides the chart recorder under the well.	on / off opens on
Readouts	The period of each ball, T₁ and T₂, in seconds.	panel
⚡ Energy ledger	KE, PE and total E for each ball, plus bars for the two balls' combined KE and PE.	panel opens closed
▶ Play / ⏸ Pause	Runs the motion, or freezes both balls where they are.	top bar
Speed	Plays the motion faster or slower on screen. The physics and the period stay the same.	0.25×–4× opens at 0.25×
↻ Reset	Restarts the mode you are in from its beginning.	top bar

Fixed: the force is always F = −kx, with no friction, so the swings never die away.

Both balls keep the same time

Release ball 1 from 0.30 m and ball 2 from 0.60 m, with k = 20 N/m and m = 0.50 kg. Ball 2 travels twice as far, yet both finish each cycle together: T₁ = T₂ = 0.993 s.

Left: the two release points and energies on the well. Right: the two traces cross zero at the same moments.

On the x(t) chart the two traces cross zero together, cycle after cycle. The amber trace is the cyan one stretched to twice the height.
This holds for any pair of amplitudes the sliders allow, and at any k and m.

Key idea. In a spring well, the time of one oscillation does not depend on how far you pull the ball before letting go.

See itRace the two balls from 0.30 m and 0.60 m. Watch for: both traces crossing the zero line at the same moments, and T₁ = T₂.

Why the amplitude cancels

Newton's second law with the spring force. The factor k/m contains no amplitude.

The spring's pull grows in proportion to distance: twice as far, twice the pull. At release, with k = 20 N/m, that is 6 N on ball 1 and 12 N on ball 2.
So ball 2 has twice the acceleration of ball 1 at every matching point of the swing. Twice the distance, covered with twice the push, takes the same time.
Its whole motion is ball 1's motion scaled up by A₂/A₁. The amplitude only multiplies the solution:

The exact motion the sim draws, for a ball released from rest at x = A.

This only works because the force is linear in x. A force that grows faster or slower than x leaves the amplitude in the period (see section 07).

See itThe same race; now look at the arrows on the x axis. Watch for: the amber arrow always twice as long as the cyan one, both shrinking to nothing at the centre.

The spring and the mass set the period

Stiffer spring, shorter period: T ∝ 1/√k. Doubling k from 20 to 40 N/m takes T from 0.993 s to 0.702 s.
Heavier ball, longer period: T ∝ √m. The spring's pull, −kx, has no m in it, so more mass means less acceleration. Four times the mass doubles T: 0.50 → 2.00 kg takes it to 1.99 s.

k m A

These sliders move the sim too.

Across the sliders' ranges, T runs from 0.44 s (k = 40 N/m, m = 0.2 kg) to 3.97 s (k = 5 N/m, m = 2 kg).

See itGive both balls four times the mass, 2.00 kg. Watch for: T₁ and T₂ both reading 1.987 s, twice 0.993 s, and still equal.

The amplitude sets the energy

Energy is conserved. The ball turns round where its energy line meets the well.

The amplitude fixes the energy, E = ½kA², and the turning points, x = ±A.
Energy grows as A²: at k = 20 N/m, ball 1 (0.30 m) has 0.90 J and ball 2 (0.60 m) has 3.60 J, four times as much.
All of it is potential energy at x = ±A and all kinetic at x = 0, where each ball is fastest.

Top speed grows with A, in step with the distance to cover.

Ball 2 crosses the centre at twice ball 1's speed (3.79 m/s against 1.90 m/s at the defaults). Twice the distance at twice the speed: the same time.

See itOpen the Energy ledger during the race. Watch for: E₂ holding at four times E₁, while T₁ and T₂ stay equal.

Where it matters

Watches. A watch's balance wheel turns against a hairspring whose torque grows with the twist. Its period stays nearly constant as the swing shrinks while the mainspring runs down.
Musical pitch. A tuning fork keeps the same pitch as its note fades: for small vibrations the period does not depend on the amplitude.
Weighing astronauts. Scales do not work in orbit. On Skylab, astronauts sat in a chair on springs and timed its oscillation; T = 2π√(m/k) then gives the mass, once the chair's own mass is taken off.

Go deeper: a well where the amplitude does matter

A pull of fixed size F₀, towards the centre, whatever the distance.

In a V-shaped well the pull does not grow with distance, so a wider swing gets no extra push and takes longer: T ∝ √A. The sim does not run this well.
With F₀ ≈ 4.86 N, a 0.30 m release matches the spring's 0.993 s, but a 0.60 m release takes √2 times longer, 1.40 s.
A pendulum is in between: its pull grows as sin θ, slower than θ, so wide swings take longer.

Same release points in both wells. Calculated, not simulated.

Summary

In one line. For a spring force F = −kx, T = 2π√(m/k) is set by the mass and the spring alone; the amplitude sets the energy, the turning points and the top speed.

Key points

Balls released from different distances in one spring well keep exactly the same time.
The pull grows in proportion to distance, so the extra push exactly pays for the extra distance.
T grows as √m and falls as 1/√k.
E = ½kA²: double the amplitude, four times the energy, twice the top speed, same period.

Common mix-ups

"A bigger swing takes longer." Not in a spring well. It is true only where the pull grows more slowly than the distance, as for a wide pendulum swing.
"A heavier ball is pulled harder, so it oscillates faster." The pull −kx has no m in it. More mass means less acceleration and a longer period.
"More energy means a slower cycle." Energy sets how far and how fast, not how long.

Things to try on your own

Set A₁ = 0.15 m and A₂ = 0.70 m, and check that the two traces still cross zero together.
Set k = 40 N/m, then find the mass that brings T back to 0.993 s.
Open the Energy ledger, pause anywhere, and compare the two balls' KE.