A transverse sinusoidal wave moves along a string in the positive $\mathrm{x}$-direction at a speed of $10 \mathrm{~cm} / \mathrm{s}$. The wavelength of the wave is $0.5 \mathrm{~m}$ and its amplitude is $10 \mathrm{~cm}$. At a particular time $t$, the snap -shot of the wave is shown in figure. The velocity of point $P$ when its displacement is $5 \mathrm{~cm}$ is Figure: $Image$
IIT 2008, Advanced
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$ \mathrm{y}=5 \mathrm{~cm} \text { and } \mathrm{V}=+\mathrm{ve} $
$ \mathrm{y}=\mathrm{A} \sin (\omega \mathrm{t} \pm \phi) \quad \mathrm{V}=\mathrm{A} \omega \cos (\omega \mathrm{t} \pm \phi)$
We get $\omega t \pm \phi=30^{\circ}$
$ \omega=2 \pi \frac{\mathrm{v}}{\lambda}=\frac{2 \pi}{5} $
$ \mathrm{v}=\mathrm{A} \omega \cos (\omega \mathrm{t}+\phi)=\left(\frac{10}{100}\right) \times\left(\frac{2 \pi}{5}\right)\left(\frac{\sqrt{3}}{2}\right)=\frac{\pi \sqrt{3}}{50} \mathrm{~m} / \mathrm{s}$
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