The amplitude of a damped oscillator decreases to $0.9$ times its original magnitude in $5\ s$. In another $10\ s$ it will decrease to $\alpha $ times its original magnitude, where $\alpha $ equals
JEE MAIN 2013, Medium
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$\because A=A_{0} e^{-\frac{b t}{2 m}}\left(\text { where }, A_{0}=\text { maximum amplitude }\right)$
$ According \,to \,the\, questions,\, after\, 5\, second, $
$0.9 \mathrm{A}_{0}=\mathrm{A}_{0} \mathrm{e}^{-\frac{\mathrm{b}(5)}{2 \mathrm{m}}}$ $...(i)$
$After\, 10 \,more\, second,$
$A=A_{0} \mathrm{e}^{-\frac{\mathrm{b}(15)}{2 \mathrm{m}}}$ $...(ii)$
From eq"s $(i)$ and $(ii)$
$A=0.729 A_{0} \therefore \alpha=0.729$
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