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 analyze the situation of damped oscillations and apply the relevant formulas.

### Step 1: Understand the Problem
We are given that after 100 oscillations, the amplitude of the oscillator reduces to \( \frac{1}{3} \) of its initial amplitude \( A_0 \). We need to find the amplitude after 200 oscillations.

### Step 2: Use the Damped Oscillation Formula
The formula for the amplitude of a damped oscillator at time \( t \) is given by:
\[ A(t) = A_0 e^{-bt} \]
where:
- \( A(t) \) is the amplitude at time \( t \),
- \( A_0 \) is the initial amplitude,
- \( b \) is the damping constant,
- \( t \) is the time or number of oscillations.

### Step 3: Set Up the Equation for 100 Oscillations
For 100 oscillations, we have:
\[ A(100) = A_0 e^{-b \cdot 100} \]
According to the problem, this amplitude is \( \frac{1}{3} A_0 \):
\[ A_0 e^{-b \cdot 100} = \frac{1}{3} A_0 \]
Dividing both sides by \( A_0 \) (assuming \( A_0 \neq 0 \)):
\[ e^{-b \cdot 100} = \frac{1}{3} \]

### Step 4: Set Up the Equation for 200 Oscillations
Now, we need to find the amplitude after 200 oscillations:
\[ A(200) = A_0 e^{-b \cdot 200} \]

### Step 5: Relate the Two Equations
We can express \( e^{-b \cdot 200} \) in terms of \( e^{-b \cdot 100} \):
\[ e^{-b \cdot 200} = (e^{-b \cdot 100})^2 \]
Substituting from our previous result:
\[ e^{-b \cdot 200} = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \]

### Step 6: Calculate the Amplitude After 200 Oscillations
Now substituting back into the amplitude equation:
\[ A(200) = A_0 e^{-b \cdot 200} = A_0 \cdot \frac{1}{9} \]

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