First slide

Damped and forced oscillations

Question

The amplitude of a damped oscillator becomes half in one minute. The amplitude after 3 minute will be $\frac{1}{X}$ times the original, where X is

Moderate

A

$2 \times 3$
B

$2^{3}$
C

$3^{2}$
D

$3 \times 2^{2}$

Solution

Amplitude of damped oscillator
$A = A_{0} e^{- λt}; λ =$ constant, t = time
For t =1 min. $\frac{A_{0}}{2} = A_{0} e^{- λ t} \Rightarrow e^{λ} = 2$
For t = 3 min. $A = A_{0} e^{- λ \times 3} = \frac{A_{0}}{{(e^{λ})}^{3}} = \frac{A_{0}}{2^{3}}$
$\Rightarrow X = 2^{3}$

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