Equation of a standing wave in a string fixed at both ends is given as y=2 A sin k x cos omega t amp

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14.Waves and Sound

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The equation of a standing wave in a string fixed at both ends is given as $y=2 A \sin k x \cos \omega t$ The amplitude and frequency of a particle vibrating at the mid of an antinode and a node are respectively

A

$A, \frac{\omega}{2 \pi}$

B

$\frac{A}{\sqrt{2}}, \frac{\omega}{2 \pi}$

C

$A, \frac{\omega}{\pi}$

D

$\sqrt{2} A, \frac{\omega}{2 \pi}$

Solution

(d)

$y=2 A \sin k x \cdot \cos \omega t$

In a standing waves the function of amplitude $\left(A_y\right)$ is given by $A_y=2 A \sin k x$

At mid-point of node and antinode $x=\frac{\lambda}{8}$

$A_y=2 A \sin \frac{2 \pi}{\lambda} \times \frac{\lambda}{8}\left[k=\frac{2 \pi}{\lambda}\right]$

$\text { or } A_y=\frac{2 A}{\sqrt{2}}$

$\therefore A_y=\sqrt{2} A$

Frequency is same at all points $=\frac{\omega}{2 \pi}$

Standard 11

Physics

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