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In damped oscillations, the amplitude of oscillations is reduced to one-third of its initial value $a_{o}$ at the end of 100 oscillations. When the oscillator completes 200 oscillations, its amplitude must be

Option 1)
$\frac{a_{0}}{}2$

Option 2)
$\frac{a_{0}}{}4$

Option 3)
$\frac{a_{0}}{}6$

Option 4)
$\frac{a_{0}}{}9$

Answers (1)

best_answer

In damped oscillation, amplitude goes on decaying exponentially
where $b =$ damping coefficient
Intially, $\frac{a_{0}}{3} = a_{o}e^{-b\times 100T}$ , $T =$ time of one oscillation

or $\frac{1}{3} = e^{-100bT}$
Finally , $a = a_{o}e^{-b\times 200T}$ or $a = a_{o}(e^{-b\times 100T})^{2}$

or $a = a_{o}(\frac{1}{3})^{2}$

or $a = a_{o}(\frac{a_{o}}{9})$

Resultant amplitude in damped oscillation -

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

$E=E_{0}.e^{-\frac{bt}{2m}}$

- wherein

$A= Amplitude$

$E= Energy$

Option 1)

$\frac{a_{0}}{}2$

This is incorrect.

Option 2)

$\frac{a_{0}}{}4$

This is incorrect.

Option 3)

$\frac{a_{0}}{}6$

This is incorrect.

Option 4)

$\frac{a_{0}}{}9$

This is correct.

Posted by

divya.saini

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