Wave Motion 2 Question 33
33. A wave disturbance in a medium is described by $y(x, t)=0.02 \cos \left(50 \pi t+\frac{\pi}{2}\right) \cos (10 \pi x)$, where $x$ and $y$ are in metre and $t$ is in second.
$(1995,2 M)$
(a) A node occurs at $x=0.15 m$
(b) An antinode occurs at $x=0.3 m$
(c) The speed of wave is $5 ms^{-1}$
(d) The wavelength of wave is $0.2 m$
Show Answer
Answer:
Correct Answer: 33. (a, b, c, d)
Solution:
- It is given that $y(x, t)=0.02 \cos (50 \pi t+\pi / 2) \cos (10 \pi x)$
$$ \cong A \cos (\omega t+\pi / 2) \cos k x $$
Node occurs when $k x=\frac{\pi}{2}, \frac{3 \pi}{2}$ etc.
$$ 10 \pi x=\frac{\pi}{2}, \frac{3 \pi}{2} \Rightarrow x=0.05 m, 0.15 m $$
option (a)
Antinode occurs when $k x=\pi, 2 \pi, 3 \pi$ etc.
$$ \begin{array}{ll} & 10 \pi x=\pi, 2 \pi, 3 \pi \text { etc. } \\ \Rightarrow \quad & x=0.1 m, 0.2 m, 0.3 m \quad \text { option (b) } \end{array} $$
Speed of the wave is given by,
$$ v=\frac{\omega}{k}=\frac{50 \pi}{10 \pi}=5 m / s $$
option (c)
Wavelength is given by,
$$ \lambda=\frac{2 \pi}{k}=\frac{2 \pi}{10 \pi}=\left(\frac{1}{5}\right) m=0.2 m $$
option (d)
