A) \[\frac{1}{5}\]
B) \[\frac{2}{3}\]
C) \[\frac{1}{6}\]
D) \[\frac{1}{9}\]
Correct Answer: D
Solution :
In case of damped vibration, amplitude at any instant t is \[a={{a}_{0}}\,{{e}^{-bt}}\] where \[{{a}_{0}}=\] initial amplitude b = damping constant Ist case: t = 100 T and a = \[\frac{{{a}_{0}}}{3}\] \[\therefore \frac{{{a}_{0}}}{3}={{a}_{0}}\,{{e}^{-b}}^{(100T)}\] \[\Rightarrow {{e}^{-100\,bT}}=\frac{1}{3}\] IInd case: t = 200 T \[a={{a}_{0}}\,{{e}^{-bt}}={{a}_{0}}\,{{e}^{-b\,(200\,T)}}\] \[={{a}_{0}}\,{{({{e}^{-100\,bT}})}^{2}}={{a}_{0}}\times {{\left( \frac{1}{3} \right)}^{2}}=\frac{{{a}_{0}}}{9}\] Thus, after 200 oscillations, amplitude will become 1/9 times.
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