A) 2a
B) 8 a
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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