Physics - Wave on String Question with Solution | TestHub

Question Explanation: The problem describes a standing wave on a long string with a known antinode at $\mathrm{x}=0$ and the next node at $\mathrm{x}=0.10\mathrm{m}$. We are also given a displacement-time (y-t) graph for the particle at the antinode ( $\mathrm{x}=$ 0 ). We need to determine which of the provided physical statements is correct. Concept: Standing Wave Equation: For an antinode at $\mathrm{x}=0$, the general equation is:

$$y(x,t)=A_{\mathrm{sw}}\cos (\mathrm{kx})\sin (\omega \mathrm{t}+\varphi )$$

Solution: Wave Parameters: The distance between an antinode and the adjacent node is $\lambda /4$. Angular frequency $(\omega ):\omega =2\pi /\mathrm{T}$. Wave number ( k ): $\mathrm{k}=2\pi /\lambda$. Velocity of string particle: ${\mathrm{v}}_{\mathrm{p}}=\frac{\partial \mathrm{y}}{\partial \mathrm{t}}$. Find Wavelength $(\lambda )$ : Distance between antinode $(\mathrm{x}=0)$ and node $(\mathrm{x}=0.10)=\lambda /4=0.10\mathrm{m}$.

$$\lambda =0.4\mathrm{m}\Rightarrow \mathrm{k}=\frac{2\pi }{0.4}=5\pi \mathrm{rad}/\mathrm{m}$$

Analyze the y -t Graph at $\mathrm{x}=0$ : From the graph, the time period $\mathrm{T}=0.4\mathrm{s}$. Frequency $\mathrm{f}=1/0.4=2.5\mathrm{Hz}$ Angular frequency $\omega =2\pi (2.5)=5\pi \mathrm{rad}/\mathrm{s}$ Amplitude at antinode ${\mathrm{A}}_{\mathrm{SW}}=2\mathrm{cm}$. Since y starts at 0 and goes negative at $\mathrm{t}=0$, the function is $-2\sin (5\pi \mathrm{t})$. Construct the Full Equation:

$$y(x,t)=-2\cos (5\pi x)\sin (5\pi t)$$

Evaluate Option $1\left(\mathrm{x}=\frac{1}{15}\mathrm{m},\mathrm{t}=0.05\mathrm{s}\right)$ :

$$\begin{array}{rlrl}& \cos \left(5\pi \cdot \frac{1}{15}\right)=\cos (\pi /3)=1/2& \sin (5\pi \cdot 0.05)=\sin (\pi /4)=1/\sqrt{2}& y=-2\cdot (1/2)\cdot (1/\sqrt{2})=-1/\sqrt{2}\mathrm{cm}\end{array}$$

This matches Option 1. Evaluate Option 3 (Wave Speed): $\mathrm{v}=\mathrm{f}\lambda =2.5\cdot 0.4=1\mathrm{m}/\mathrm{s}$ (Option 3 says 2 m/s, so it is incorrect). Final Answer: Option (1)