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:
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:
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Updated On: Mar 18, 2026
- \( \frac{1}{2e} \)
- \( \frac{2}{e} \)
- \( \frac{1}{e^2} \)
- \( \frac{2}{e^2} \)
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The Correct Option is C
Approach Solution - 1
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: \[ \boxed{\frac{1}{e^2} A_0} \]
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Approach Solution -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).
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