If a simple pendulum has significant amplitude (up to a factor of 1/e… | JEE Main 2012 (AIEEE)

Q23JEE Main 2012 · AIEEEDifficult

Damped, Forced Oscillations & Superposition

If a simple pendulum has significant amplitude (up to a factor of $1/e$ of original) only in the period between $t = 0\text{ s to } t = \tau\text{ s},$ then $\tau$ may be called the average life of the pendulum When the spherical bob of the pendulum suffers a retardation (due to viscous drag) proportional to its velocity with $b$ as the constant of proportionality, the average life time of the pendulum is (assuming damping is small) in seconds:

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Answer: (d)

Retardation $= bv$ , so drag force $= mbv$ and $m\ddot{x} + mb\dot{x} + kx = 0$ .
Amplitude $A = A_0 e^{-bt/2}$ .
$A = A_0/e$ when $\frac{b\tau}{2} = 1 \Rightarrow \tau = \frac{2}{b}$ .
So the answer is (d) $\frac{2}{b}$ .

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Damped, Forced Oscillations & Superposition question from JEE Main 2012 (AIEEE)

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