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A simple pendulum is said to have significant amplitude only between t = 0 and t = tau s, where tau is the 'average life' (amplitude falls to 1/e of its original value). The spherical bob experiences a viscous retardation proportional to its velocity, with proportionality constant b (i.e. the damping term is b*v). Assuming light damping, find the average lifetime tau (in seconds).

1. 2/b
2. 1/b
3. 0.693/b
4. b

Correct answer: 2/b

Solution

For damped oscillation m x'' + b x' + k x = 0, the amplitude decays as A(t) = A0 e^(-(b/2m) t). The amplitude falls to 1/e (i.e. average life) when (b/2m) tau = 1, giving tau = 2m/b. With the retardation given simply as 'b times velocity' per unit mass (m taken as 1 as in the standard JEE version of this problem), tau = 2/b.

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