An audio transmitter (T) and receiver (R) hang from identical strings of length 8 m, pulled in oppos

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Question

An audio transmitter (T) and receiver (R) hang from identical strings of length 8 m, pulled in opposite directions by small angular amplitude θ₀ = cos⁻¹(0.9) and released simultaneously. Natural frequency of transmitter = 660 Hz, speed of sound = 330 m/s, g = 10 m/s². The maximum variation in frequency (in Hz) as measured by the receiver is

Accepted Answer

Correct Answer: 32.01 HzStep-by-step Solution:Step 1: Find maximum velocity of each pendulum.Energy conservation for pendulum: (1/2)mv² = mgL(1 − cosθ₀)v_max = √(2gL(1−cosθ₀)) = √(2 × 10 × 8 × (1−0.9)) = √(2 × 10 × 8 × 0.1) = √16 = 4 m/sStep 2: Apply Doppler effect.When T and R approach each other (both moving toward each other simultaneously), the maximum apparent frequency:f_max = f₀·(v + v_R)/(v − v_T) = 660·(330 + 4)/(330 − 4) = 660·334/326= 660·334/326 = 220·334/326/1... let me compute: 660 × 334 = 220440. 220440/326 ≈ 676.2 Hz.When T and R move away from each other (both at maximum velocity in opposite directions):f_min = f₀·(v − v_R)/(v + v_T) = 660·(330 − 4)/(330 + 4) = 660·326/334= 215160/334 ≈ 644.2 Hz.Maximum variation = f_max − f_min = 676.2 − 644.2 = 32.0 Hz ≈ 32.01 HzKey Concepts:Pendulum velocity: v_max = √(2gL(1−cosθ)) obtained from energy conservation. At θ₀ = cos⁻¹(0.9): 1−cosθ₀ = 1−0.9 = 0.1, giving v_max = 4 m/s.Doppler effect general formula: f_obs = f_source × (v_sound + v_observer)/(v_sound − v_source). Signs: + for approach, − for recession (for both observer and source velocities). When both approach: v_R → +v_R, v_T → +v_T. When both recede: both become negative.Maximum frequency variation in oscillating source-observer system: The maximum Δf occurs when source and observer move with maximum speed simultaneously (which happens at the equilibrium position for pendulums). Maximum frequency: both moving toward each other. Minimum frequency: both moving away.Simple pendulum period and velocity: T = 2π√(L/g). The velocity at position θ: v = √(2gL(cosθ − cosθ₀)). Maximum at θ = 0 (equilibrium): v_max = √(2gL(1−cosθ₀)).

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