First slide

Damped oscillation

Question

The amplitude of a damped oscillator becomes half in one minute. The amplitude after 3 minutes will be 1/x times the initial amplitude, where x is

Moderate

A

6
B

8
C

9
D

12

Solution

$\begin{array}{l} A_{t} = A_{0} e^{- kt} where k = constant \\ ∴ \frac{A_{0}}{2} = A_{0} e^{- kt} \Rightarrow e^{- kt} = \frac{1}{2} \Rightarrow e^{- k} = \frac{1}{2} [∴ t = 1 \sec] \\ Again \frac{A_{0}}{x} = A_{0} e^{- 3 k} \Rightarrow \frac{1}{x} = {(\frac{1}{2})}^{3} \Rightarrow x = 8 \end{array}$

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