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: \[t=200\text{ }T\] |
| \[a={{a}_{0}}\,{{e}^{-bt}}={{a}_{0}}\,{{e}^{-b\,(200\,T)}}\] |
| \[={{a}_{0}}\,{{({{e}^{-100\,bT}})}^{2}}={{a}_{0}}\times {{\left( \frac{1}{3} \right)}^{2}}=\frac{{{a}_{0}}}{9}\] |
| Thus, after 200 oscillations, amplitude will become 1/9 times. |
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