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Question

If a simple pendulum has a significant amplitude (up to a factor of 1/e of original) only in the period between t=0 s to t=τ s, then τ 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 it's 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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Solution

For damped harmonic motion
ma=kxmbv
or,ma+mbv+kx=0
solution of above equation is
x=A0edfracbt2sinωtwithω2=kmb24
where, amplitude drops exponentially with time.
Aτ=A0ebt2
Average time r is that duration when amplitude drops by 63% i,e., becomes A0e.
Aτ=A0e=A0ebt2
bt2=1orτ=2b

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