A) \[{{3}^{2}}\]
B) \[{{3}^{3}}\]
C) \[\sqrt[3]{3}\]
D) \[{{2}^{3}}\]
Correct Answer: B
Solution :
[b] Amplitude of a damped oscillator at any instant t is given by \[A={{A}_{0}}{{e}^{-bt/2m}}\] where\[{{A}_{0}}\] is the original amplitude From question, When \[t=2s,\,A=\frac{{{A}_{0}}}{3}\,\,\therefore \,\frac{{{A}_{0}}}{3}={{A}_{0}}{{e}^{-2b/2m}}\] or \[\frac{1}{3}={{e}^{-b/m}}\] ?. (i) When \[t=6s,\,A=\frac{{{A}_{0}}}{n}\] \[\therefore \,\frac{{{A}_{0}}}{n}={{A}_{0}}{{e}^{-6b/2m}}\] or,\[\frac{1}{n}={{e}^{-3b/m}}={{({{e}^{-b/m}})}^{3}}\] or, \[\frac{1}{n}={{\left( \frac{1}{3} \right)}^{3}}\] \[\therefore \,n={{3}^{3}}\] (Using eq. (i))
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