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(B) For a string clamped at both ends,the length $\ell$ in the $n^{th}$ harmonic is given by $\ell = n \frac{\lambda}{2}$.Given $n = 4$,we have $\ell

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$80$

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(B) For a string clamped at both ends,the length $\ell$ in the $n^{th}$ harmonic is given by $\ell = n \frac{\lambda}{2}$.Given $n = 4$,we have $\ell = 4 \frac{\lambda}{2} = 2\lambda$.The general equation of a stationary wave is $Y = A \sin(kx) \cos(\omega t)$,where $k = \frac{2\pi}{\lambda}$.Comparing the given equation $Y = 0.3 \sin(0.157 x) \cos(200\pi t)$ with the general form,we get $k = 0.157$.Since $0.157 \approx \frac{\pi}{20}$,we have $\frac{2\pi}{\lambda} = \frac{\pi}{20}$.Solving for $\lambda$,we get $\lambda = 40 \, m$.Substituting this into the length formula: $\ell = 2 \lambda = 2 	imes 40 = 80 \, m$.

$A$ string is clamped at both ends and it is vibrating in its $4^{th}$ harmonic. The equation of the stationary wave is $Y = 0.3 \sin(0.157 x) \cos(200\pi t)$. The length of the string is ..... $m$ (all quantities are in $SI$ units).

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