An undamped oscillator has period ro = 1 second. When weak

Step 1 of 4

The expression for the period of an undamped oscillator in terms of the natural frequency of the undamped oscillator is,

\( {\tau _o} = \frac{{2\pi }}{{{\omega _o}}} \)

The period of the undamped oscillator is \({\tau _o} = 1\;{\rm{s}}\). Thus, the equation becomes as:

\(\frac{{2\pi }}{{{\omega _o}}} = 1\;{\rm{s}}\;\;\;\;...\;\left( 1 \right)\)

Here, \({\omega _o}\) is the natural frequency of the undamped oscillator.

The expression for the period of the damped oscillator in terms of the damped oscillator is,

\({\tau _1} = \frac{{2\pi }}{{{\omega _1}}}\;\;\;\;\;\;\;...\;\left( 2 \right)\)

Here, \({\omega _1}\) is the frequency of the damped oscillator.

The frequency of the damped oscillator is,

\({\omega _1} = \sqrt {\omega _o^2 - {\beta ^2}} \)

Here, \(\beta\) is the damping constant.

Substitute \(\sqrt {\omega _o^2 - {\beta ^2}}\)  for \({\omega _1}\) in the equation (2).

\({\tau _1} = \frac{{2\pi }}{{\sqrt {\omega _o^2 - {\beta ^2}} }}\;\;\;\;\;\;\;\;...\;\left( 3 \right)\)