🔄 Lesson 8: SHM & Phasors

Simple harmonic motion as rotating vectors — the elegant view of oscillations

🎯 Learning Objectives

• Understand phasor representation of SHM
• Connect position, velocity, acceleration phasors
• Use phasors to add oscillations and solve interference problems
• Apply energy conservation in SHM using phasor geometry

🔑 Key Concepts

What Is a Phasor?

A phasor is a rotating vector of length A (amplitude) that rotates at constant angular velocity ω. Its projection on the x-axis gives x(t) = A cos(ωt + φ).

SHM = Uniform Circular Motion viewed from the side!

Phasor Relationships

Quantity	Expression	Phasor Length	Phase vs Position
Position	x = A cos(ωt + φ)	A	0
Velocity	v = -Aω sin(ωt + φ) = Aω cos(ωt + φ + π/2)	Aω	leads by 90° (π/2)
Acceleration	a = -Aω² cos(ωt + φ) = Aω² cos(ωt + φ + π)	Aω²	leads by 180° (π)

Energy in Phasor View

Total energy E = ½kA² = ½mω²A² = ½mv_max² = ½kx_max²

At any instant: KE = ½mv² = ½mA²ω² sin²(ωt + φ), PE = ½kx² = ½mω²A² cos²(ωt + φ)

KE + PE = constant = ½mω²A² — the phasor tip moves on a circle, energy partitions between projections!

Adding SHM Oscillations (Same Frequency)

x₁ = A₁ cos(ωt), x₂ = A₂ cos(ωt + φ)

Resultant x = A cos(ωt + δ) where:

A² = A₁² + A₂² + 2A₁A₂ cos φ
tan δ = (A₂ sin φ) / (A₁ + A₂ cos φ)
            

This is just vector addition of phasors!

📐 Worked Examples

Example 1: SHM Phasors

Mass-spring: m=0.5kg, k=50N/m, A=0.1m. Find ω, v_max, a_max, E.

ω = √(k/m) = √(50/0.5) = √100 = 10 rad/s
v_max = Aω = 0.1 × 10 = 1 m/s
a_max = Aω² = 0.1 × 100 = 10 m/s²
E = ½kA² = ½(50)(0.01) = 0.25 J
Check: ½mv_max² = ½(0.5)(1) = 0.25 J ✓
            

Example 2: Phasor Addition — Two Oscillators

Two identical oscillators (A=2cm, ω=5 rad/s) with phase difference φ=60°. Find resultant.

A₁ = A₂ = 2 cm
A² = 2² + 2² + 2(2)(2)cos60° = 4 + 4 + 8(½) = 12
A = √12 = 2√3 ≈ 3.46 cm
tan δ = (2 sin60°)/(2 + 2 cos60°) = (2×√3/2)/(2+1) = √3/3 = 1/√3
δ = 30° = π/6
x = 3.46 cos(5t + π/6)
            

Example 3: Phase Between Position, Velocity, Acceleration

At what phase(s) is KE = PE? What about KE = 3×PE?

KE = PE  →  ½mv² = ½kx²  →  ω²A² sin²θ = ω²A² cos²θ  (unless A=0)
sin²θ = cos²θ  →  tan²θ = 1  →  θ = 45°, 135°, 225°, 315° (π/4, 3π/4, 5π/4, 7π/4)

KE = 3 PE  →  sin²θ = 3 cos²θ  →  tan²θ = 3  →  tanθ = ±√3
θ = 60°, 120°, 240°, 300°
            

🏋️ Practice Problems

1. SHM: x = 0.2 cos(8t). Find A, ω, T, f, v_max, a_max.
2. m=2kg, k=200N/m, A=0.05m. Find total energy, v_max, ω.
3. Two oscillations: x₁ = 3 cos(ωt), x₂ = 4 cos(ωt + 90°). Find resultant A, δ.
4. In SHM, at what displacement is speed half of maximum? (x/A = ?)
5. Two pendulums: L₁=1m, L₂=4m. Same ω? Ratio of periods? Phase after 10s if started together?
6. Phasor of length 5 rotates at ω=3 rad/s. At t=0, angle=30°. Find x, v, a at t=1s.

✅ Answers

Click to reveal answers

1) A=0.2, ω=8, T=π/4, f=2/π, v_max=1.6, a_max=12.8 2) E=0.25J, v_max=0.5m/s, ω=10rad/s 3) A=5, δ=53.1° 4) x = A√3/2 ≈ 0.866A 5) T₂=2T₁, after 10s: pend1 ~5 cycles, pend2 ~2.5 cycles → out of phase 6) x≈2.24, v≈-12.7, a≈-38.2

🎉 Trigonometry Review Complete!

You've mastered the trigonometry foundation for Physics 101. The skills you've practiced — right triangle trig, unit circle, vector components, wave superposition, and phasors — will be used throughout the course, especially in:

• 2D Kinematics (projectile motion, relative velocity)
• Forces (inclines, tension, equilibrium)
• Waves & Sound (interference, beats, standing waves)
• Oscillations (SHM, energy, damping, resonance)

You're ready for Physics 101!

🚀 Start Physics 101 →