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=$\tau$ 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 it's velocity, with b as the constant of proportionality, the average life time of the pendulum is (assuming damping is small) in seconds

Answer 1

For damped harmonic motion

$ma=-kx-mbv$

$or,ma+mbv+kx=0$

solution of above equation is

$x=A_0{e}^{dfrac-bt2}\sin \omega twith{\omega }^2=\frac{k}{m}-\frac{b^2}{4}$

where$,$ amplitude drops exponentially with time$.$

$A_{\tau }=A_0{e}^{\frac{-bt}{2}}$

Average time $r$ is that duration when amplitude drops by $63\%$ $i,e.,$ becomes $\frac{A_0}{e}.$

$A_{\tau }=\frac{A_0}{e}=A_0{e}^{\frac{-bt}{2}}$

$\frac{bt}{2}=1or\tau =\frac{2}{b}$