A) \[\frac{0.693}{b}\]
B) b
C) \[\frac{1}{b}\]
D) \[\frac{2}{b}\]
Correct Answer: D
Solution :
[d] For damped harmonic motion, \[ma=-kx-mbv\] or \[ma+mbv+kx=0\] Solution to above equation is \[x={{A}_{0}}{{e}^{-\frac{bt}{2}}}\,\sin \,\omega t;\] with \[{{\omega }^{2}}=\frac{k}{m}-\frac{{{b}^{2}}}{4}\] where amplitude drops exponentially with time i.e., \[{{A}_{\tau }}={{A}_{0}}{{e}^{-\frac{b\tau }{2}}}\] Average time \[\tau \] is that duration when amplitude drops by 63%, i.e., becomes\[{{A}_{0}}/e\]. Thus, \[{{A}_{\tau }}=\frac{{{A}_{0}}}{e}={{A}_{0}}{{e}^{-\frac{b\tau }{2}}}\] or \[\frac{b\tau }{2}=1\,\,\,or\,\,\,\tau =\frac{2}{b}\]You need to login to perform this action.
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