First slide

Damped and forced oscillations

Question

The amplitude of a damped oscillations decreases to 0.9 times its initial value in 5 seconds. By how many times to its initial value, energy of oscillation decreases in first 10 seconds?

Moderate

A

0.81
B

0.73
C

0.95
D

0.66

Solution

Amplitude of damped oscillator is given by $A = A_{0} e^{- \frac{b t}{2 m}}$ in 5 second
$\begin{array}{l} 0.9 A_{0} = A_{0} e^{- \frac{5 b}{2 m}} \\ In 10 second , enregy is given by E = E_{0} e^{- \frac{bt}{m}} \\ E = E_{0} e^{- \frac{10 b}{m}} \\ \Rightarrow E = E_{0} {(e^{- \frac{5 b}{m}})}^{2} = E_{0} {({(0.9)}^{2})}^{2} = 0.656 E_{0} \\ ⇒Energy \approx 0.66 E_{0} \end{array}$

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