Question

In a time of 2 s, the amplitude of a damped oscillator becomes \( \frac{1}{e} \) times its initial amplitude \( A \). In the next two seconds, the amplitude of the oscillator is:

• A. \( \frac{1}{2e} \)
• B. \( \frac{2}{e} \)
• C. \( \frac{1}{e^2} \)
• D. \( \frac{2}{e^2} \)

Hint:

N/A

Answer 1

The Correct Option is C

Step 1: General Equation for Damped Amplitude

The amplitude \( A(t) \) of a damped oscillator is given by: \[ A(t) = A_0 e^{-\gamma t} \] Where:

• \( A_0 \) is the initial amplitude,
• \( \gamma \) is the damping constant,
• \( t \) is the time in seconds.

 

Step 2: Apply the Given Condition

After 2 seconds, it's given: \[ A(2) = \frac{1}{e} A_0 = A_0 e^{-2\gamma} \] Divide both sides by \( A_0 \): \[ \frac{1}{e} = e^{-2\gamma} \] Take natural log on both sides: \[ -1 = -2\gamma \quad \Rightarrow \quad \gamma = \frac{1}{2} \]

Step 3: Calculate Amplitude After 4 Seconds

\[ A(4) = A_0 e^{-\gamma \cdot 4} = A_0 e^{-4 \cdot \frac{1}{2}} = A_0 e^{-2} \] \[ A(4) = \frac{1}{e^2} A_0 \]

Conclusion:

The amplitude of the oscillator after 4 seconds is: \[ \boxed{\frac{1}{e^2} A_0} \]

Answer 2

We are given the following information: - In 2 seconds, the amplitude of a damped oscillator becomes \( \frac{1}{e} \) times its initial amplitude \( A \). - We are asked to find the amplitude of the oscillator in the next 2 seconds. The equation for the amplitude \( A(t) \) of a damped oscillator is given by: \[ A(t) = A_0 e^{-\gamma t}, \] where \( A_0 \) is the initial amplitude, \( \gamma \) is the damping coefficient, and \( t \) is the time. ### Step 1: Finding the damping coefficient In the first 2 seconds, the amplitude decreases to \( \frac{A}{e} \), so: \[ \frac{A}{e} = A_0 e^{-\gamma \cdot 2}. \] This simplifies to: \[ \frac{1}{e} = e^{-2\gamma}, \] which gives: \[ 2\gamma = 1 \quad \Rightarrow \quad \gamma = \frac{1}{2}. \] ### Step 2: Amplitude after the next 2 seconds In the next 2 seconds, the time will be 4 seconds in total. The amplitude at \( t = 4 \) seconds is: \[ A(4) = A_0 e^{-\gamma \cdot 4}. \] Substitute \( \gamma = \frac{1}{2} \): \[ A(4) = A_0 e^{-\frac{4}{2}} = A_0 e^{-2}. \] Thus, the amplitude of the oscillator after the next 2 seconds will be: \[ A(4) = \frac{A_0}{e^2}. \] Therefore, the amplitude of the oscillator after the next 2 seconds is \( \frac{1}{e^2} \). Thus, the correct answer is option (3).