Question

A wave disturbance in a medium is described by $y(x,t)=0.02cos⟮50\pi t+\frac{\pi }{2}⟯cos\left(10\pi x\right)$, where x and y are in metres and t in seconds

• A. A displacement node occurs at x = 0.15 m
• B. An antinode occurs at x = 0.3 m
• C. The wavelength of the wave is 0.2 m
• D. The speed of the wave is 5.0 m/s

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A

A displacement node occurs at x = 0.15 m

B

An antinode occurs at x = 0.3 m

C

The wavelength of the wave is 0.2 m

D

The speed of the wave is 5.0 m/s

$y=0.02cos\left(10\pi x\right)cos⟮50\pi t+\frac{\pi }{2}⟯$

At node, amplitude = 0

$\Rightarrow cos\left(10\pi x\right)=0\Rightarrow 10\pi x=\frac{\pi }{2},\frac{3\pi }{2}$

$\Rightarrow x=\frac{1}{20}=0.05m,0.15m.......$

At antinode, amplitude is maximum

$\Rightarrow cos\left(10\pi x\right)=\pm 1\Rightarrow x=0,\pi ,2\pi ....$

$\Rightarrow x=0,0.1m,0.2m....$

Now $\lambda =2\times \text{Distance between two nodes or antinodes}$

$=2\times 0.1=0.2m$ and $\frac{2\pi vt}{\lambda }=50\pi t$

$v=25\lambda =25\times 0.2=5m/sec$.