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When an oscillator completes 100 oscillations its amplitude reduces to $\dfrac{1}{3}$ of initial value. What will be its amplitude, when it completes 200 oscillations : -
A
$\dfrac{1}{8}$
B
$\dfrac{2}{3}$
C
$\dfrac{1}{6}$
D
$\dfrac{1}{9}$
Detailed Solution
For damped oscillations the amplitude decays exponentially: $A = A_0e^{-bt}$.
After 100 oscillations ($t = 100T$): $\dfrac{A_0}{3} = A_0e^{-b(100T)} \Rightarrow e^{-100bT} = \dfrac{1}{3}$
After 200 oscillations ($t = 200T$): $A = A_0e^{-b(200T)} = A_0(e^{-100bT})^2$
$\Rightarrow A = A_0\left(\dfrac{1}{3}\right)^2 = \dfrac{A_0}{9}$
So the amplitude becomes $\dfrac{1}{9}$ of the initial value. (In equal time intervals the amplitude falls by the same factor, here $\dfrac{1}{3}$ each time.)
After 100 oscillations ($t = 100T$): $\dfrac{A_0}{3} = A_0e^{-b(100T)} \Rightarrow e^{-100bT} = \dfrac{1}{3}$
After 200 oscillations ($t = 200T$): $A = A_0e^{-b(200T)} = A_0(e^{-100bT})^2$
$\Rightarrow A = A_0\left(\dfrac{1}{3}\right)^2 = \dfrac{A_0}{9}$
So the amplitude becomes $\dfrac{1}{9}$ of the initial value. (In equal time intervals the amplitude falls by the same factor, here $\dfrac{1}{3}$ each time.)