Waves - Result Question 14
14. A transverse wave is represented by the equation $y=y_0 \sin \frac{2 \pi}{\lambda}(v t-x)$
For what value of $\lambda$ is the maximum particle velocity equal to two times the wave velocity?
[1998]
(a) $\lambda=2 \pi y_0$
(b) $\lambda=\frac{\pi y_0}{3}$
(c) $\lambda=\frac{\pi y_0}{2}$
(d) $\lambda=\pi y_0$
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Answer:
Correct Answer: 14. (d)
Solution:
- (d) $y=y_0 \sin \frac{2 \pi}{\lambda}(v t-x)$
Particle velocity,
$\frac{d y}{d t}=y_0 \times \frac{2 \pi}{\lambda} v \cos \frac{2 \pi}{\lambda}(v t-x)$
Maximum particle velocity $=y_0 \times \frac{2 \pi \nu}{\lambda}$
Wave velocity $=v \quad$ [given]
So, $y_0 \times \frac{2 \pi v}{\lambda}=2 v$
$\lambda=\pi \cdot y_0$
