When an oscillator completes `100` oscillations its amplitude reduced to `(1)/(3)` of initial values. What will be amplitude, when it completes `200` oscillations :

Options

A. `1/8`
B. `2/3`
C. `1/6`
D. `1/9`

Explanation

To solve the problem step by step, we will use the concept of damped harmonic motion. The amplitude of a damped oscillator decreases exponentially over time.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know that after 100 oscillations, the amplitude \( A \) of the oscillator is reduced to \( \frac{1}{3} \) of its initial value \( A_0 \).

2. **Amplitude Reduction Formula**:
The amplitude of a damped oscillator at any time \( t \) can be expressed as:
\[ A(t) = A_0 e^{-bt} \]
where \( b \) is a damping constant.

3. **Time for 100 Oscillations**:
Let \( T_1 \) be the time taken to complete 100 oscillations. Thus, we can express the amplitude after 100 oscillations as:
\[ A(100) = A_0 e^{-bT_1} = \frac{A_0}{3} \]

4. **Setting Up the Equation**:
From the equation above, we can rearrange to find:
\[ e^{-bT_1} = \frac{1}{3} \]

5. **Time for 200 Oscillations**:
The time taken to complete 200 oscillations is \( T_2 = 2T_1 \). We can express the amplitude after 200 oscillations as:
\[ A(200) = A_0 e^{-bT_2} = A_0 e^{-b(2T_1)} = A_0 (e^{-bT_1})^2 \]

6. **Substituting the Value**:
We know from step 4 that \( e^{-bT_1} = \frac{1}{3} \). Therefore:
\[ A(200) = A_0 \left(\frac{1}{3}\right)^2 = A_0 \cdot \frac{1}{9} \]

7. **Final Result**:
Thus, the amplitude after 200 oscillations is:
\[ A(200) = \frac{A_0}{9} \]

### Conclusion:
The amplitude when the oscillator completes 200 oscillations is \( \frac{A_0}{9} \).