Question Detail
Question
An oscillator has a period of 3s and its amplitude decreases by 5% each cycle. What is the time constant (τ) for this oscillator?
Correct Answer
Option A is the correct answer.
Detailed Explanation
From Worked Example 33 (page 39), the amplitude decreases by 5% each cycle means A/A₀ = 0.95 after one period T. Using the amplitude decay formula A = A₀e^(-T/2τ), we have 0.95 = e^(-3/2τ). Taking the natural logarithm: ln(0.95) = -3/(2τ). So, τ = -3 / (2 * ln(0.95)) = -3 / (2 * (-0.05129)) ≈ 29.24 s. This matches the calculation in the example.
Hint
Use the formula for amplitude decay: A = A₀e^(-t/2τ). The amplitude decreases by 5% after one period (T). So, A/A₀ = 0.95 when t = T.