Section 6 — Perpendicular SHM Superposition & Lissajous Figures
GE-4 / PHYS7021 unit 6 · 3 lecture-hours · 2 solved questions
6.1 Theory you need
x = A sin ωt, y = A sin(ωt+φ) ⇒ x2/A2 + y2/A2 − 2xy cos φ/A2 = sin2φ (ellipse in general)- φ = 0 → straight line y = x; φ = π/2 → circle x2+y2 = A2; φ = π → line y = −x.
- Different frequencies (ratio m:n) → Lissajous figures; the pattern is stable when the ratio is exact.
Uses: (1) comparing two frequencies (ratio from number of tangencies); (2) measuring phase difference; (3) testing signal generators; (4) displaying an unknown frequency on a CRO against a known one.
6.2 Solved past questions — exam-style answers
1. What are Lissajous figures? (two applications / electrical demonstration)
- Definition: when a particle (e.g., an electron spot on a CRO) executes two perpendicular SHMs whose frequencies are in a simple ratio, the closed stationary curve traced on the screen is called a Lissajous figure; its shape depends on the frequency ratio and the phase difference.
- Electrical demonstration: feed two a.c. signals from two oscillators into the X- and Y-plates of a CRO; a stable closed pattern appears when the ratio is exact.
- Applications: (i) frequency comparison — ratio = (horizontal tangencies)/(vertical tangencies); (ii) phase measurement between two signals of the same frequency.
∴ Closed CRO patterns from perpendicular SHMs; used to compare f and φ
2. Two SHMs of same frequency & amplitude, phase difference π/2, perpendicular — shape of resultant motion?
Given: x = A sin ωt, y = A sin(ωt + π/2) = A cos ωt.
- Square and add: x2 + y2 = A2(sin2ωt + cos2ωt) = A2.
- This is the equation of a circle of radius A; the particle moves uniformly round it.
∴ A circle of radius A