A wave in a string has an amplitude of $2 \mathrm{~cm}$. The wave travels in the +ve direction of…

A wave in a string has an amplitude of $2 \mathrm{~cm}$. The wave travels in the +ve direction of $\mathrm{x}$-axis with a speed of $128 \mathrm{~m} / \mathrm{s}$ and it is noted that 5 complete waves fit in $4 \mathrm{~m}$ length of the string. The equation describing the wave is
  1. $y=(0.02) m \sin (7.58 x-1005 t)$
  2. $y=(0.02) \mathrm{m} \sin (7.85 x+1005 \mathrm{t})$
  3. $y=(0.02) m \sin (15.7 x-2010 t)$
  4. $y=(0.02) m \sin (15.7 x+2010 t)$

Solution

Key Idea Find the parameters and put in the general wave equation. Here, \(\begin{aligned} A & =2 \mathrm{~cm} \\ \text {direction } & =+\mathrm{ve} \text { direction } \\ \mathrm{v} & =128 \mathrm{~ms}^{-1} \end{aligned}\) and \(5 \lambda=4\) Now, \(\mathrm{k}=\frac{2 \pi}{\lambda}=\frac{2 \pi \times 5}{4}=7.85\) \(\begin{aligned} &\text {and }\\ &\mathrm{v}=\frac{\omega}{\mathrm{k}}=128 \mathrm{~ms}^{-1}\\ & \Rightarrow \omega=\mathrm{v} \times \mathrm{k}=128 \times 7.85 \\ & =1005 \\ & \text {As, } y=\mathrm{A} \sin (\mathrm{kx}-\omega \mathrm{t}) \\ & \therefore y=2 \sin (7.85 x-1005 t) \\ & =(0.02) \mathrm{m} \sin (7.85 \mathrm{x}-1005 \mathrm{t}) \end{aligned}\)

Asked in: NEET 2009 (Mains)

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