SHM = projection of circular motion
A peg on a turntable (radius r, angular speed ω) casts a shadow on a screen: θ = ωt, shadow displacement x = r sin ωt — SHM with amplitude r and the same ω. Shadow speed passing the centre = rω; shadow acceleration when instantaneously at rest (edges) = rω². This is why ω, rad s⁻¹ and phase-as-angle appear in a straight-line motion.
What this actually means
Spin a peg on a turntable and light it from the side: the shadow on the wall performs exact SHM with amplitude r and the turntable's ω.
This is where all the angular language of SHM comes from. Phase is literally the angle the peg has swept; 2π of angle is one full cycle of the shadow.
Useful numbers transfer directly: shadow speed through the centre is rω (the peg's full speed, momentarily parallel to the screen) and shadow acceleration at the edges is rω² (the peg's centripetal acceleration, momentarily along the screen).
Prove it — watch it be true
- Watch the turntable peg and its shadow side by side
- The shadow oscillates in a straight line while the peg circles
- STEP π/4 and watch angle swept translate directly into phase gained