Circular motion ↔ SHM

SHM is the projection of uniform circular motion — v leads x by π/2, a antiphase with x.

peg on turntable → shadow on screen → strip records x(t)

drag to orbit · scroll to zoom

Phasor view

rotating vectors · vertical projections give x, v, a · lengths normalised

v-phasora-phasor

time t

0.00 s

phase ωt + φ (mod 2π)

0.00 rad · 0°

first revolution

x = x₀ sin(ωt + φ)

+0.00 m

v = ωx₀ cos(ωt + φ)

+1.40 m/s

a = −ω²x₀ sin(ωt + φ)

+0.00 m/s²

phasor magnitudes

x₀ 0.70 · ωx₀ 1.40

ω²x₀ 2.80

Try first: Pause, then press Step +π/4 four times.

Angular frequency ω2.0 rad/s

T = 2π/ω = 3.14 s · f = 0.32 Hz

Amplitude x₀ (= radius)0.70 m

turntable radius = shadow amplitude

Phase offset φ+0.00 rad (0°)

shifts where the motion starts in its cycle

Start convention

switching restarts at t = 0

each Step = π/4 = T/8

What to notice

SHM is a projection. Uniform circular motion at angular speed ω, viewed edge-on (the shadow), IS simple harmonic motion with the same ω. The radius of the circle is the amplitude x₀.
v leads x by π/2. The shadow is fastest sweeping through equilibrium (the peg moves parallel to the screen) and momentarily at rest at ±x₀ (the peg moves along the beam). So v peaks a quarter period before x — its phasor sits 90° ahead.
a is antiphase with x. a leads v by another π/2, which puts it 180° from x: whenever x is positive, a is equally negative. That is the geometric face of a = −ω²x — the defining equation of SHM.
One revolution = one period. Each full turn of the peg is exactly one oscillation of the shadow: 2π rad of phase per cycle. A quarter turn = T/4 = a 90° phase step — check it against the T/4 gridlines.