A stationary wave of amplitude A is generated between the two fixed ends x = 0 and x = L. The particle at x = L/3 is a node. There are only two particles between x = L/6 and x = L/3 which have maximum speed half of the maximum speed of the anti-node. Again there are only two particles between x = 0 and x = L/6 which have maximum speed half of that at the antinodes. The slope of the wave function at x = L/3 changes with respect to time according to the graph shown. The symbols $μ, ω$ and A are having their usual meanings if used in calculations.

The time period of oscillations of a particle is

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T/2

answer is B.

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Detailed Solution

$(\begin{array}{lll} x = 0 is a Node \\ \frac{\partial y}{\partial x} = 0 at t = 0 \end{array}) \Rightarrow y = Asin kx, \sin ωt$

$\frac{\partial y}{\partial x} = Ak \cdot \cos kx \cdot \sin ωt$

Wavelength, $λ = \frac{L}{3} = \frac{2 π}{k}$

$\Rightarrow k = \frac{6 π}{L}$

$Thus, \frac{\partial y}{\partial x} = \frac{6 Aπ}{L} \cdot \sin ωt$

Question Image

$ω . 2 T = 2 π$

$\frac{2 π}{T'} . 2 T = 2 π$

$\Rightarrow T' = \frac{2 π}{ω} = 2 T$

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