Superposition, Phasors, and Lissajous Motion
Linear SHMs of common angular frequency superpose to another SHM whose complex amplitude is the vector sum; perpendicular components may instead trace a Lissajous curve.
Why this shows up in the exam
Resultant amplitude and phase · Cancellation of oscillations · Identifying circular, elliptical, or Lissajous paths
Learn the idea
Same-frequency SHMs add as rotating vectors, while perpendicular components generate geometric trajectories. Represent each sinusoid by an arrow rotating at the common angular speed; the vector sum immediately gives the resultant amplitude and phase.
🧠 Memory hook: Same frequency: add arrows, not amplitudes.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- R² = A² + B² + 2AB cos(delta) — resultant amplitude for two collinear equal-frequency SHMs
- Z = sum_j A_j exp(i phi_j) — phasor sum for multiple components
- x = A sin(at + delta), y = B sin(bt) — parametric form of a Lissajous trajectory
How to approach it
- 1Check whether frequencies match
- 2Draw or compute the phase vectors
- 3For two axes, eliminate time or inspect the frequency ratio
Common slip-ups that cost marks
- •Adding amplitudes arithmetically for nonzero phase
- •Using phasors for unequal frequencies without qualification
- •Ignoring the direction of perpendicular components
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A mass of 1 kg is attached to a spring of force constant 100 N/m. Find the angular frequency of small oscillations.
More from Oscillations and Waves
Superposition, Beats, and Resonance
Explore the principles of superposition, formation of beats, beat frequency, resonance, and phase difference in wave phenomena.
Standing Waves and Harmonics
Understand the formation of standing waves, harmonics in strings and organ pipes (open and closed), and the relationship between displacement and pressure in stationary waves.
Velocity and Acceleration in SHM
Explore how velocity and acceleration vary with displacement and time in SHM, including maximum values and their physical significance.
Simple Pendulum and Spring Systems
Study the time period formulas, energy considerations, and behavior of simple pendulums and spring-mass systems, including effects of non-inertial frames and spring combinations.
Basics of Simple Harmonic Motion (SHM)
Understand the defining features of SHM, including its standard equations, conditions, and the relationship between displacement, velocity, and acceleration.
Wave Motion and Wave Equation
Learn the fundamentals of wave motion, including the wave equation, its parameters, direction, and the distinction between particle and wave velocities.