Oscillations Ques 6
6. The equation of a simple harmonic wave is given by
$ y=3 \sin \frac{\pi}{2}(50 t-x) $
Where $x$ and $y$ are in meters and $t$ is in seconds. The ratio of maximum particle velocity to the wave velocity is
[2012 M]
(a) $2 $ $\pi$
(b) $\frac{3}{2} $ $\pi$
(c) $3$ $ \pi$
(d) $\frac{2}{3}$ $ \pi$
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Answer:
Correct Answer: 6.(b)
Solution: (b) $y=3 \sin \frac{\pi}{2}(50 t-x)$
$y=3 \sin \left(25 \pi t-\frac{\pi}{2} x\right)$ on comparing with the standard wave equation
$y=a \sin (\omega t-k x)$
Wave velocity $v=\frac{\omega}{k}=\frac{25 \pi}{\pi / 2}=50 \mathrm{m} / \mathrm{sec}$.
The velocity of particle
$ \begin{aligned} & v_p=\frac{\partial y}{\partial t}=75 \pi \cos \left(25 \pi t-\frac{\pi}{2} x\right) \\ & v_{p \max }=75 \pi \\ & \text { then } \frac{v_{p_{\max }}}{v}=\frac{75 \pi}{50}=\frac{3 \pi}{2} \end{aligned} $
BETA
