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(B) The given equation is $y(x, t) = (0.01 \ m) \sin(62.8x) \cos(628t)$.$(A)$ For the $n^{th}$ harmonic,the number of loops is $n = 5$. The number of

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$(B, C)$

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(B) The given equation is $y(x, t) = (0.01 \ m) \sin(62.8x) \cos(628t)$.$(A)$ For the $n^{th}$ harmonic,the number of loops is $n = 5$. The number of nodes is $n + 1 = 5 + 1 = 6$. Thus,statement $(A)$ is incorrect.$(B)$ The wave number $k = 62.8 \ m^{-1}$. Since $k = \frac{2\pi}{\lambda}$,we have $\lambda = \frac{2 	imes 3.14}{62.8} = 0.1 \ m$. For the $5^{th}$ harmonic,the length of the string $L = \frac{5\lambda}{2} = \frac{5 	imes 0.1}{2} = 0.25 \ m$. Thus,statement $(B)$ is correct.$(C)$ The amplitude of the standing wave is $A(x) = 0.01 \sin(62.8x)$. At the midpoint $x = \frac{L}{2} = 0.125 \ m$,$A(0.125) = 0.01 \sin(62.8 	imes 0.125) = 0.01 \sin(7.85) \approx 0.01 \sin(2.5\pi) = 0.01 	imes 1 = 0.01 \ m$. Thus,statement $(C)$ is correct.$(D)$ The angular frequency $\omega = 628 \ rad/s$. The frequency $f = \frac{\omega}{2\pi} = \frac{628}{2 	imes 3.14} = 100 \ Hz$. The fundamental frequency $f_1 = \frac{f}{n} = \frac{100}{5} = 20 \ Hz$. Thus,statement $(D)$ is incorrect.Therefore,the correct statements are $(B)$ and $(C)$.

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