How this instrument works
Simple harmonic motion is what happens when something is pulled back toward a center by a force proportional to how far it has moved — a mass on a spring stretched by Hooke's law, a pendulum swinging through a small angle, the tine of a tuning fork. That restoring force produces a particular differential equation, acceleration equal to −ω² times displacement, and the only well-behaved solution to it is a sinusoid. For an object released from its extreme position, that solution is x = A·cos(ωt): amplitude A is the extreme displacement, fixed by however far the thing was pulled before release, while angular frequency ω is set by the physics of the system itself and never depends on how hard it was started.
The three formulas here are one differentiation apart. Velocity is the rate of change of displacement, dx/dt; acceleration is the rate of change of velocity, dv/dt. Differentiating the cosine turns it into a negative sine for velocity, then back into a negative cosine, scaled by ω², for acceleration. That is why acceleration always equals −ω² times displacement, pointed back toward the center, strongest exactly where the object is momentarily still and weakest where it is moving fastest — velocity and displacement are a quarter cycle out of step, trading kinetic motion for stored restoring force and back.
The model assumes the restoring force stays perfectly proportional to displacement and that nothing drains energy from the system: a spring that never fatigues, a pendulum swinging only a few degrees, no friction or air resistance. Swing a pendulum past roughly 15° and the small-angle approximation that makes it harmonic starts to measurably break down; leave real friction in a spring and the amplitude decays cycle by cycle, turning the clean cosine into a slowly dying spiral. This instrument returns the ideal, undamped case — the first approximation every mechanics course and most quick engineering checks start from before adding damping terms.
- Enter the Amplitude — the maximum displacement from center, in metres or another length unit from the menu.
- Enter the Angular frequency in radians per second; for a system oscillating at frequency f, that is ω = 2πf.
- Set the Time for the snapshot you want — t = 0 is the moment the clock starts, typically the instant of release.
- Read Displacement, Velocity, and Acceleration together: position, speed, and restoring push at that exact instant.
- Change Time and recompute to step through a cycle and watch the three quantities trade their peaks and zeros.
Worked example — released from full extension
Pull a mass on a spring 10 cm off center and let it go: amplitude A = 0.1 m, starting from rest — the standard 'released from full extension' condition, like a pendulum bob pulled aside and let go. Suppose the system completes one full cycle per second, f = 1 Hz, so angular frequency is ω = 2πf = 6.283185307 rad/s. The snapshot is taken at the instant of release, t = 0 s, and the instrument evaluates all three formulas at that instant.
Displacement: x = 0.1 × cos(6.283185307 × 0) = 0.1 × 1 = 0.1 m — exactly the full amplitude, right where the mass was let go. Velocity: v = −0.1 × 6.283185307 × sin(0) = −0.1 × 6.283185307 × 0 = 0 m/s — momentarily still, as anything is at a turning point. Acceleration: a = −0.1 × 6.283185307² × cos(0) = −0.1 × 39.4784176... = −3.94784176 m/s² — the full restoring push, pulling the mass back toward center as hard as it will ever pull during this cycle.
Questions
What is angular frequency, and how does it differ from ordinary frequency?
Angular frequency ω, in radians per second, equals 2π times ordinary frequency f, in cycles per second (hertz): ω = 2πf. A 1 Hz oscillator, one full cycle per second, has ω = 6.283185307 rad/s, because a single cycle sweeps through 2π radians. The formulas here use ω directly because differentiating a cosine of ωt is cleaner in radians than in cycles; convert first if a plain frequency is what you were given.
Why is displacement written as a cosine and not a sine?
Cosine describes an object released from its point of maximum displacement — a mass pulled aside and let go, which is the starting condition this instrument assumes. At t = 0, cos(0) = 1, so x = A, matching that release point exactly. An object instead starting at center and moving at its fastest is described by a sine, x = A sin(ωt); the two curves are identical, only shifted by a quarter cycle, depending on where the clock starts.
How are velocity and displacement related during the motion?
They are a quarter cycle apart and never both large at once. Velocity peaks at Aω right at the center, where displacement is zero, and drops to zero at the two extremes, ±A, where displacement peaks. Energy simply trades between kinetic, moving fast through the center, and potential, stopped at full stretch, which is why v = −Aω·sin(ωt) hits zero exactly when x = A·cos(ωt) hits its amplitude.
Does a larger amplitude make the oscillation faster?
No — that independence is what makes the motion 'simple.' Angular frequency ω comes entirely from the system's physical properties, spring stiffness over mass (ω = √(k/m)) or gravity over pendulum length (ω = √(g/L)), never from how far it was pulled before release. Pull a spring 2 cm or 20 cm and it still completes each cycle in the same time; amplitude changes how far it swings, not how fast it cycles.
What happens to acceleration as the object crosses the center?
It passes through zero. Acceleration is a = −Aω²·cos(ωt), and cos(ωt) is zero exactly when the object crosses center, so the restoring force momentarily vanishes there as well. Nothing is pulling it back at that instant, which is exactly why inertia carries it straight through to the other side rather than stopping it at the middle.
Is real-world oscillation ever truly simple harmonic motion?
Only approximately. Real springs stiffen or fatigue away from their rest length, real pendulums stay harmonic only for small swing angles, and friction or air resistance drains energy this idealized model ignores, so real amplitude slowly decays instead of repeating forever. Engineers still start with the undamped equations here because they capture the dominant behavior — period, peak speed, peak force — with three easily measured numbers before any damping term is added.