Phase Relationships in SHM
Overview
Starting from x = A·cos(ωt): Displacement: x = A·cos(ωt). Velocity: v = −Aω·sin(ωt). Acceleration: a = −Aω²·cos(ωt) = −ω²x. Phase differences: v leads x by 90° (π/2 rad) — when x is at maximum, v = 0; when x = 0, v is maximum. a is 180° (π rad) out of phase with x — when x is maximum positive, a is maximum negative. This is why acceleration always points back toward equilibrium (restoring). Energy: KE = ½mω²(A² − x²), PE = ½mω²x², Total E = ½mω²A² = ½kA² = constant.
Phase and phase difference
Phase φ describes the position in the oscillation cycle. If x = A cos(ωt + φ₀), then φ₀ is the initial phase (at t = 0). Phase difference Δφ between two oscillators describes how far apart they are in their cycles. If Δφ = 0: in phase (they move together). If Δφ = π: antiphase (one at +A when the other at −A). If Δφ = π/2: 90° out of phase. Phase is measured in radians.
- ⚠Mixing up degrees and radians for phase — always use radians in SHM formulas
- ⚠Forgetting that a phase difference of 2π means the oscillators ARE in phase (they repeat every 2π)