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  4. In damped oscillations, the amplitude of oscillations is reduced to one-third of its initial value a0 at the end of 100 oscillations. When the oscillator completes 200 oscillations, its amplitude must be

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Q. In damped oscillations, the amplitude of oscillations is reduced to one-third of its initial value $ {{a}_{0}} $ at the end of $ 100 $ oscillations. When the oscillator completes $ 200 $ oscillations, its amplitude must be

KEAMKEAM 2007Oscillations

Solution:

In damped oscillation, amplitude goes on decaying exponentially
$ a={{a}_{0}}{{e}^{-bt}} $
where $ b= $ damping coefficient.
Initially, $ \frac{{{a}_{0}}}{3}={{a}_{0}}{{e}^{-b\times 100T}} $
where T is time of one oscillation.
Or $ \frac{1}{3}={{e}^{-100bT}} $ .. (i)
Finally, $ a={{a}_{0}}{{e}^{-b\times 200T}} $
Or $ a={{a}_{0}}{{[{{e}^{-100bT}}]}^{2}} $
Or $ a={{a}_{0}}\times {{\left[ \frac{1}{3} \right]}^{2}} $ [from Eq.(i)]
Or $ a=\frac{{{a}_{0}}}{9} $