How to FIND φ — the recipe
(1) Read T off either graph. (2) Pick the same feature on both curves — peak→peak, or zero-crossings going the same direction. (3) Measure the time offset Δt between them. (4) Convert fraction of cycle to angle. The curve whose feature comes earlier leads. From equations instead: φ = difference of the brackets — sin vs cos with the same argument is automatically π/2. Traps: up-crossing vs down-crossing silently adds π; take the nearest corresponding feature — if φ > π, quote 2π − φ and swap leads↔lags; φ = 0 in phase, φ = π antiphase.
φ = (Δt/T) × 2π rad = (Δt/T) × 360°What this actually means
The recipe: read T; pick the SAME feature on both curves (peak to peak, or zero crossings heading the same way); measure the time offset Δt; convert with (Δt/T) × 2π. The curve whose feature arrives earlier leads.
'Going the same direction' is the silent killer: matching an up-crossing on one curve with a down-crossing on the other adds a hidden π to your answer.
Take the nearest matching feature. If you land on φ greater than π, the neater answer is 2π − φ with leads and lags swapped.
From equations, φ is simply the difference of the brackets, and sin against cos of the same argument is automatically π/2.
Matching an up-crossing with a down-crossing — it silently adds π.
Prove it — watch it be true
- Put the x and v curves side by side and read T
- Measure peak-to-peak offset: Δt = T/4
- (T/4)/T × 2π = π/2, and v's peak comes earlier: v leads x