The equation of a stationary wave on a string | vedclass

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(D) The standard equation of a standing wave is $y = 2A \sin(kx) \cos(\omega t)$.Comparing the given equation $Y = 0.5 \sin(0.314 x) \cos(600 \pi t)$

Vedclass · Accepted Answer

$30$

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(D) The standard equation of a standing wave is $y = 2A \sin(kx) \cos(\omega t)$.Comparing the given equation $Y = 0.5 \sin(0.314 x) \cos(600 \pi t)$ with the standard form,we identify the wave number $k = 0.314 \ cm^{-1}$.Since $0.314 \approx \frac{\pi}{10}$,we have $k = \frac{\pi}{10} \ cm^{-1}$.The wavelength $\lambda$ is given by $k = \frac{2\pi}{\lambda}$,so $\frac{\pi}{10} = \frac{2\pi}{\lambda}$,which gives $\lambda = 20 \ cm$.For a string clamped at both ends,the length $L$ for the $n^{th}$ harmonic is $L = n \frac{\lambda}{2}$.For the third harmonic,$n = 3$,so $L = 3 	imes \frac{20}{2} = 3 	imes 10 = 30 \ cm$.

The equation of a stationary wave on a string clamped at both ends and vibrating in the third harmonic is $Y = 0.5 \sin(0.314 x) \cos(600 \pi t)$,where $x$ and $y$ are in $cm$ and $t$ is in seconds. The length of the vibrating string is: (in $cm$)

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