When an oscillator completes $100$ oscillations its amplitude reduced to $ \frac{1}{3} $ of initial value. What will be its amplitude, when it completes $200$ oscillations?

Question

What is the fundamental frequency of an open organ pipe with length $ L $ and speed of sound $ v $?

Answer 1

To solve this problem, we need to understand how the amplitude of an oscillator decays over time, particularly in a damped oscillator. This is usually described by an exponential decay of amplitude.

Given that the amplitude reduces to $\frac{1}{3}$ of its initial value after 100 oscillations, we can express this exponential decay as:

A_1 = A_0 \cdot e^{-bN}

where:

A_1 = \frac{1}{3} A_0 is the amplitude after 100 oscillations.
A_0 is the initial amplitude.
b is the damping factor per oscillation.
N = 100 is the number of oscillations.

Thus, we can write:

\frac{1}{3} A_0 = A_0 \cdot e^{-100b}

On simplifying, we get:

e^{-100b} = \frac{1}{3}

Next, we need to find the amplitude after 200 oscillations (A_2):

A_2 = A_0 \cdot e^{-200b}

Substituting e^{-100b} = \frac{1}{3} in the above equation:

e^{-200b} = (e^{-100b})^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9}

Therefore, the amplitude after 200 oscillations is:

A_2 = A_0 \cdot \frac{1}{9} = \frac{1}{9} A_0

Thus, the amplitude of the oscillator after 200 oscillations is $\frac{1}{9}$ of the initial value.

The correct answer is therefore:

$\frac{1}{9}$

Answer 2