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APEX CLASS ESTD 2018

IITJEE: SHM Phasor Circle, Energy Parabolas & Organ Pipe Harmonics

By

Ashish Kumar Bahekar

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Apex Class • IITJEE SHM & Standing Waves Lab

SHM Phasor Circle, Energy Parabolas & Organ Pipe Harmonics

Master SHM Time Shortcuts (T/12 vs. T/6), Spring Cutting Rules, and Open vs. Closed Organ Pipe Overtones!

🔬 Select an SHM Concept to Inspect Phasor Angles, Energy & Spring Constants:
+A/2 +A 30° 60° Phasor Circle (ω = 2π / T) 🏆 Instant SHM Time-Interval Cheat Table 1. Mean (x = 0) → Half-Amplitude (x = A/2): sin(ωt₁) = 1/2 ⇒ ωt₁ = π/6 (30°) ⇒ t₁ = T / 12! 2. Half-Amplitude (x = A/2) → Extreme (x = A): Δθ = 90° – 30° = 60° (π/3) ⇒ t₂ = T / 6 (Twice as long!) 3. Mean (x = 0) → x = A/√2 (45°): t = T / 8 ⚡ Why t₂ = 2 t₁? Because the particle slows down as it approaches the extreme point (v = 0 at x = A)! Total Energy E = ½ k A² (Constant) +A/√2 -A/√2 -A +A SHM Energy Checkpoints • Where is K = U? At x = ± A / √2! • At x = A/2: U = ¼ E and K = ¾ E! • Time-Average over 1 Full Cycle: ⟨K⟩_time = ⟨U⟩_time = ¼ k A² = ½ E • Frequency of K & U = 2 × f_SHM! 🎯 Most Repeated JEE Trap: If SHM displacement frequency is f, Kinetic & Potential Energy oscillate at frequency 2f! v_max = +ωA +A v² / (ωA)² + x² / A² = 1 (Ellipse!) a = -ω² x Slope = -ω² Phase Diff: v leads x by 90°; a leads x by 180° 🌀 Velocity at any position x: v = ω √(A² – x²) | Maximum Acceleration at extreme ends: a_max = ω² A! 1. Series vs. Parallel Spring Combinations • Parallel Springs (Same Extension x): k_eq = k₁ + k₂ ⇒ T = 2π √[ m / (k₁ + k₂) ] • Series Springs (Same Tension F): 1/k_eq = 1/k₁ + 1/k₂ ⇒ k_eq = k₁k₂ / (k₁ + k₂) (Notice Springs work like Capacitors, opposite to Resistors!) 2. Cutting a Spring Trap (k · L = Constant!) • Stiffness is inversely proportional to length: k₁ L₁ = k₂ L₂ = k L • Cut into n equal pieces: Each has k’ = n · k! • Cut in ratio 1 : 2 (Lengths L/3 & 2L/3): k₁ = 3k (Short piece) & k₂ = 1.5k (Long piece)! 🔩 Two-Block Reduced Mass Trick: If m₁ and m₂ are connected by spring k on a smooth floor, T = 2π √(μ / k) where μ = m₁m₂/(m₁+m₂)!
Spring-Block Period
T = 2π √(m / k)
Independent of Gravity g & Incline θ!
Simple Pendulum Period
T = 2π √(L / g_eff)
In Lift going up: g_eff = g + a
Second’s Pendulum
T = 2.0 sec | L ≈ 1.0 m
Infinite Length Pendulum: T = 84.6 min
Physical / Rod Pendulum
T = 2π √[ I / (mg d) ]
Uniform Rod pivoted at end: L_eq = ⅔L
🎵 Standing Wave Node-to-Node Distance = λ/2, Node-to-Antinode = λ/4. Select System:
n=1 (Fundamental / 1st Harmonic): f₁ = v / (2L) n=2 (1st Overtone / 2nd Harmonic): f₂ = 2 f₁ n=3 (2nd Overtone / 3rd Harmonic): f₃ = 3 f₁ 🎸 String Wave Speed: v = √(T / μ) where μ = Mass/Length = ρA ⇒ Fundamental f₁ = [1 / (2L)] √(T / μ)! A N A 1st Harmonic (Fundamental): f₁ = v / (2L) [L = λ/2] 2nd Harmonic (1st Overtone): f₂ = 2v / (2L) = 2 f₁ 3 Nodes & 4 Antinodes Inside Pipe (L = 3λ / 2) 3rd Harmonic (2nd Overtone): f₃ = 3v / (2L) = 3 f₁ 🎺 Open Organ Pipe Rule: Produces ALL integer harmonics (f₁ : f₂ : f₃ = 1 : 2 : 3), so mᵗʰ Overtone = (m + 1)ᵗʰ Harmonic! N A Fundamental (1st Harmonic): f₁ = v / (4L) [L = λ/4] 1st Overtone = 3rd Harmonic! f₃ = 3v / (4L) = 3 f₁ L = 5λ / 4 (3 Nodes & 3 Antinodes) 2nd Overtone = 5th Harmonic! f₅ = 5v / (4L) = 5 f₁ ⚠️ #1 JEE Organ Pipe Trap: Closed Pipe has ONLY ODD Harmonics (1 : 3 : 5 : 7), so mᵗʰ Overtone = (2m + 1)ᵗʰ Harmonic! 1. End Correction (e = 0.6 r) at Open Ends Antinode forms slightly outside the pipe mouth by e = 0.6 r: • Closed Pipe (1 Open End): L_eff = L + 0.6 r • Open Pipe (2 Open Ends): L_eff = L + 1.2 r • Pressure Node = Displacement Antinode! 2. Resonance Tube Lab Manual Formulas • 1st Resonance Length: L₁ + e = λ / 4 • 2nd Resonance Length: L₂ + e = 3λ / 4 • Speed of Sound: v = 2 f (L₂ – L₁) • End Correction: e = (L₂ – 3L₁) / 2 🔬 Experimental Physics Shortcut: Subtracting L₁ from L₂ cancels end correction ‘e’ completely → λ = 2(L₂ – L₁)!
Fundamental Frequency
f₁ = v / (2L) f_open = v / (2L) f_closed = v / (4L) v = 2 f (L₂ – L₁)
Notice f_open = 2 × f_closed!
Harmonic Ratio
1 : 2 : 3 : 4 (All) 1 : 2 : 3 : 4 (All) 1 : 3 : 5 : 7 (Odd Only!) L₂ ≈ 3 L₁
Allowed Resonance Modes
Speed of Sound in Gas
v = √(γ R T / M)
Laplace Correction (v ∝ √T)
Beats Frequency
f_beat = |f₁ – f₂|
Waxing fork lowers f; Filing raises f
🔥 2 High-Yield JEE Wave Shortcuts:
1. Open Pipe Dipped Halfway in Water: It turns into a Closed Pipe of half length (L’ = L/2), so its new fundamental frequency is f’ = v / [4(L/2)] = v / (2L) = UNCHANGED!
2. Travelling Wave Speed from Equation y = A sin(ωt – kx): Wave speed is always v_wave = ω / k, while maximum particle speed is v_p,max = ω A!
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