A phase clock
Geometric reference, not a particle's orbit
One point, through time
A fixed four-second window
- Period T
- Frequency f
- Angular frequency ω
- Current value u(t)
- Elapsed cycles ft
- Accumulated phase ωt
- Initial phase φ
- Total phase θ = ωt + φ
- Circle angle α in [0, 2π)
- Arc length s = rα
- Radian ratio s/r
- Projection cos θ
The circle angle removes full turns for the geometric arc. Total phase above retains them. Changing the reference radius changes s, but not s/r, phase, amplitude or the time history.
Five landmarks in one period
Measured from t = 0 for the current phase| Event | t (s) | Phase (rad) | u (cm) | Motion | Select event |
|---|
For a zero-phase cosine these are +A, 0, −A, 0, +A. With another initial phase the quarter-period times stay the same, but the values change. At equilibrium the point may be rising or falling: displacement alone does not identify the full state of the cycle.
Building the sinusoid
One cycle becomes 2π radians
Radian measure compares an arc's length s with its circle's radius r: α = s/r. A circumference has length 2πr, so a full turn is 2π radians, whatever radius you choose. The gold arc shows the current angle within a single turn. The teal arrow turns counterclockwise as positive time increases.
The rotating arrow is a mathematical phase reference. The marked coral point represents a single oscillating quantity. In cosine form, the circle's horizontal projection gives u/A = cos θ; in sine form, the vertical projection gives u/A = sin θ. Neither picture claims that a physical oscillator moves around a circle.
Three ways to describe the same rate
One period T corresponds to one cycle, so f = 1/T counts cycles per second. The same cycle advances by 2π radians, giving ω = 2πf = 2π/T. A 1 Hz oscillation advances by 2π rad/s, not 1 rad/s. The three rate-entry modes describe the same setting.
The history keeps the same time and displacement scales as you change parameters. Doubling f halves T and doubles ω without changing A. In the optional doubled-frequency comparison, both curves use the same amplitude, initial phase and trigonometric convention.
Amplitude and the chosen time origin
In u(t) = A cos(ωt + φ), A scales the range of values, ω controls the rate of repetition, and φ selects where the cycle starts at t = 0. For φ = 0, a cosine starts at +A; for φ = π it starts at −A. For φ = ±π/2 it starts at equilibrium, but moves in opposite directions.
Changing φ changes the physical time history relative to the selected origin. Switching sine and cosine is different: the lab adds or subtracts π/2 from the phase constant so that the physical motion is unchanged. The reference arrow and selected projection axis then change together. All trigonometric calculations use radians.
The article's two worked examples
Example 1 uses A = 4.0 cm and f = 2.0 Hz, starting at the positive maximum. Thus ω = 4π rad/s, T = 0.50 s, and u(t) = 4.0 cos(4πt) cm. Quarter-period steps visit +4, 0, −4, 0 and +4 cm.
Example 2 uses A = 6.0 mm, T = 0.40 s and φ = π/2. Thus f = 2.5 Hz and ω = 5π rad/s. At t = 0 the displacement is zero and is decreasing. The preset changes display units to millimeters; changing units yourself never changes the underlying motion.
WM02 describes a time history at one point. No wavelength, propagation, forces, energy or numerical differential-equation solution is implied. At A = 0 the quantity is constant: the clock can still show nominal generator settings, but there is no observable oscillation phase or smallest positive period. The bead and history keep a fixed ±5 cm range (±50 mm); the reference circle has its own geometric radius.