A) \[\frac{0.693}{b}\]
B) b
C) \[\frac{1}{b}\]
D) \[\frac{2}{b}\]
Correct Answer: D
Solution :
\[m\frac{{{d}^{2}}x}{d{{t}^{2}}}=-kx-b\frac{dx}{dt}\] \[m\frac{{{d}^{2}}x}{d{{t}^{2}}}+b\frac{dx}{dt}+kx=0\]here b is demping coefficient This has solution of type \[x={{e}^{\lambda t}}\] substituting this \[m{{\lambda }^{2}}+b{{\lambda }^{2}}+k=0\] \[\lambda =\frac{-b\pm \sqrt{{{b}^{2}}-4mk}}{2m}\] on solving for \[x\], we get \[x={{e}^{-\frac{b}{2m}t}}\] \[a\,\cos \,({{\omega }_{1}}\,t-\alpha )\] \[{{\omega }_{1}}=\sqrt{\omega _{0}^{2}-{{\lambda }^{2}}}\] where \[{{\omega }_{0}}=\sqrt{\frac{k}{m}}\] \[\lambda =+\frac{b}{2}\] So, average life \[=\frac{2}{b}\]
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