Spacing between two successive nodes in a sta | vedclass

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(B) The spacing between two successive nodes in a standing wave is given by $d = \frac{\lambda}{2} = x$.From the wave equation,$V = f\lambda$,where $V

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$\sqrt{2}x$

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(B) The spacing between two successive nodes in a standing wave is given by $d = \frac{\lambda}{2} = x$.From the wave equation,$V = f\lambda$,where $V$ is the wave speed,$f$ is the frequency,and $\lambda$ is the wavelength.The speed of a wave on a string is $V = \sqrt{\frac{T}{\mu}}$,where $T$ is the tension and $\mu$ is the linear mass density.Since $f$ is constant,$\lambda = \frac{V}{f} \propto V \propto \sqrt{T}$.Let the initial tension be $T$ and the new tension be $T' = 2T$.The new wavelength $\lambda'$ is given by $\frac{\lambda'}{\lambda} = \sqrt{\frac{T'}{T}} = \sqrt{\frac{2T}{T}} = \sqrt{2}$.Thus,$\lambda' = \sqrt{2}\lambda$.The new spacing between nodes is $d' = \frac{\lambda'}{2} = \sqrt{2} \left( \frac{\lambda}{2} \right) = \sqrt{2}x$.

Spacing between two successive nodes in a standing wave on a string is $x$. If the frequency of the standing wave is kept unchanged but the tension in the string is doubled,then the new spacing between successive nodes will become

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