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

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

Solution

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

Asked in: NEET 2009 (Screening)

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