Answer
See the detailed answer below.
Work Step by Step
a) $T$ is the period which is the time needed for an oscillation system to complete one full cycle. And $\tau$ is the damping time constant which is the time needed for the amplitude of a damped oscillator to decrease by $e^{-1}$ from its original value.
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b) $\tau$ is the damping time constant which is the time needed for the amplitude of a damped oscillator to decrease by $e^{-1}$ from its original value, while $t_{1/2}$ is the half-life time which is the time needed for the amplitude of a damped oscillator to decrease by $50\%$ from its original value.
