Question

The displacement of a damped oscillator is \( x(t) = \exp(-0.2t) \cos(3.2t + \phi) \), where \( t \) is time in seconds. The time required for the amplitude of the oscillator to become \( \frac{1}{e^{1.2}} \) times its initial amplitude is

• A. 3 s
• B. 6 s
• C. 2 s
• D. 8 s

Hint:

For damped oscillators, use the exponential decay of the amplitude to solve for the time when it reaches a specific fraction of its initial value.

Answer 1

The Correct Option is B

Step 1: The amplitude of the damped oscillator is given by the exponential term in the displacement equation: \[ A(t) = A_0 \exp(-0.2t) \] where \( A_0 \) is the initial amplitude. 

Step 2: We are asked to find the time when the amplitude becomes \( \frac{1}{e^{1.2}} \) times its initial amplitude. This means: \[ A(t) = \frac{A_0}{e^{1.2}} \] Substitute the expression for \( A(t) \): \[ A_0 \exp(-0.2t) = \frac{A_0}{e^{1.2}} \] 

Step 3: Cancel \( A_0 \) from both sides: \[ \exp(-0.2t) = \frac{1}{e^{1.2}} \] Taking the natural logarithm of both sides: \[ -0.2t = -1.2 \] \[ t = \frac{-1.2}{-0.2} = 6 \, \text{s} \] Thus, the time required for the amplitude to become \( \frac{1}{e^{1.2}} \) times its initial amplitude is 6 s.