Question

If the amplitude of oscillation of a damped harmonic oscillation falls to 1/e of its initial value then the time taken by the oscillator is………………..​

Answer 1

Given: Let initial amplitude be A

          Final amplitude of damped oscillation = $\frac{1}{e}$A

To Find: Time taken by oscillation to fall from initial to final value

Solution:

• Formula used for damped oscillations is:

                      A = $A_{0}$$e^{-\frac{bt}{2m} }$                  (1)

Where, A is current amplitude

            $A_{0}$ is initial amplitude

            e is base of natural logarithm

            b is coefficient of damping

            t is time taken

    and m is mass of oscillator

Substituting values in (1)

                      ⇒  $\frac{A_{0} }{e}$ = $A_{0}$$e^{\frac{-bt}{2m} }$

                     ⇒    $\frac{1}{e}$ = $e^{\frac{-bt}{2m} }$

                     ⇒   $e^{-1}$ = $e^{\frac{-bt}{2m} }$

                     ⇒    $\frac{bt}{2m}$ = 1                              (equating powers of e)

                     ⇒   t = $\frac{2m}{b}$s

Therefore, it takes t = $\frac{2m}{b}$s for oscillator's amplitude to fall to $\frac{1}{e}$ of its initial value.