G L Mittal and TARUN MITTAL Solutions for Chapter: Free, Damped and Forced Oscillations, Exercise 2: FOR DIFFERENT COMPETITIVE EXAMINATIONS

If a simple pendulum has significant amplitude (up to a factor of $\frac{1}{e}$ of original) only in the period between $t=0$ to$\mathit{ }t=\mathrm{\tau }$, then $\mathrm{\tau }$ 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 proportionality, the average lifetime of the pendulum in (assuming damping is small) in seconds$:$

The amplitude of a damped oscillator decreases to $0.9$ times its original magnitude in $5 \mathrm{s}$. In another $10 \mathrm{s}$ it will decrease to $\alpha$- times its original magnitude, where $\mathrm{\alpha }$ equals$:$