A) \[2a\]
B) \[8a\]
C) \[4a\]
D) \[a\]
Correct Answer: B
Solution :
The energy \[(E)\] of a wave of amplitude \[a\], and angular velocity \[\omega \] is \[E=\frac{1}{2}m{{a}^{2}}{{\omega }^{2}}\] Also, \[\omega =2\pi n\] \[\therefore \] \[E=\frac{1}{2}m{{a}^{2}}{{(2\pi n)}^{2}}=2m\,\,{{a}^{2}}{{\pi }^{2}}{{n}^{2}}\] \[\therefore \] \[\frac{{{E}_{A}}}{{{E}_{B}}}=\frac{{{({{a}_{A}}{{n}_{A}})}^{2}}}{{{({{a}_{B}}{{n}_{B}})}^{2}}}\] Given, \[{{E}_{A}}={{E}_{B}},\,\,{{n}_{A}}=n,\,\,{{n}_{B}}=\frac{n}{8}\] \[\therefore \] \[1=\frac{a_{A}^{2}\times 64{{n}^{2}}}{a_{B}^{2}{{n}^{2}}}\] \[\Rightarrow \] \[{{a}_{B}}=8{{a}_{A}}=8a\]
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