A) \[\sqrt{2}\pi \]
B) \[16\sqrt{2}\pi \]
C) \[24\sqrt{2}\pi \]
D) \[\frac{4}{\pi }\]
E) \[\frac{32\sqrt{2}}{\pi }\]
Correct Answer: E
Solution :
For simple harmonic motion, \[y=a\sin \omega t\] \[\therefore \] \[y=a\sin \left( \frac{2\pi }{T} \right)t\] \[(at\,t=2s)\] \[{{y}_{1}}=a\sin \left[ \left( \frac{2\pi }{16} \right)\times 2 \right]\] \[=a\sin \left( \frac{\pi }{4} \right)=\frac{a}{\sqrt{2}}\] ?(i) At\[t=4\text{ }s\]or after 2 s from mean position, \[{{y}_{1}}=\frac{a}{\sqrt{2}},velocity=4\,m{{s}^{-1}}\] \[\therefore \] \[Velocity=\omega \sqrt{{{a}^{2}}-y_{1}^{2}}\] or \[4=\left( \frac{2\pi }{16} \right)\sqrt{{{a}^{2}}-\frac{{{a}^{2}}}{2}}\] [from Eq. (i)] or \[4=\frac{\pi }{8}\times \frac{a}{\sqrt{2}}\] or \[a=\frac{32\sqrt{2}}{\pi m}\]
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