Question

The amplitude of a damped harmonic oscillator becomes \( \frac{1}{n} \) times its initial amplitude \( A_0 \) at the end of 20 oscillations. The amplitude of the oscillator when it completes 40 oscillations is:

• A. \( \frac{A_0}{n^3} \)
• B. \( A_0 \)
• C. \( \frac{A_0}{n^2} \)
• D. \( \frac{A_0}{n} \)

Hint:

The damping factor in an oscillatory system affects the amplitude, and it decreases exponentially over time.

Answer 1

The Correct Option is C

- The amplitude of a damped harmonic oscillator reduces exponentially with time. If the amplitude becomes \( \frac{1}{n} \) of its initial value after \( N \) oscillations, the relation is: \[ A = \frac{A_0}{n^N} \]
- Given that after 20 oscillations, the amplitude is \( \frac{1}{n} A_0 \), we can calculate the amplitude after 40 oscillations: \[ A = \frac{A_0}{n^2} \]