a
The equation of motion for the pendulum, suffering retardation

\(I \alpha=-m g(\ell \sin \theta)-m b v(\ell)\) where \(I=m \ell^{2}\)

and \(\alpha=\mathrm{d}^{2} \theta / d t^{2}\)

\(\therefore \frac{d^{2} \theta}{d t^{2}}=-\frac{g}{\ell} \tan \theta+\frac{b v}{\ell}\)

On solving we get \(\theta=\theta_{0} e^{-\frac{b t}{2} \sin (\omega t+\phi)}\)

According to questions \(\frac{\theta_{0}}{e}=\theta_{0} e^{\frac{-b \tau}{2}}\)

\(\therefore \tau=\frac{2}{b}\)