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प्रश्न
The equation of a standing wave, produced on a string fixed at both ends, is
\[y = \left( 0 \cdot 4 cm \right) \sin \left[ \left( 0 \cdot 314 {cm}^{- 1} \right) x \right] \cos \left[ \left( 600\pi s^{- 1} \right) t \right]\]
What could be the smallest length of the string?
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उत्तर
Given:
Equation of the standing wave:
\[y = \left( 0 . 4 cm \right) \sin \left[ \left( 0 . 314 {cm}^{- 1} \right) x \right]\cos \left[ \left( 600 \pi s^{- 1} \right) t \right]\]
\[ \Rightarrow k = 0 . 314 = \frac{\pi}{10}\]
\[Also, k = \frac{2\pi}{\lambda}\]
\[ \Rightarrow \lambda = 20 \text{ cm }\]
We know:
\[L = \frac{n\lambda}{2}\]
For the smallest length, putting n = 1:
\[\Rightarrow L = \frac{\lambda}{2} = \frac{20 \text{ cm}}{2} = 10 \text{ cm }\]
Therefore, the required length of the string is 10 cm.
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