A string has mass per unit length of $10^{-6} \mathrm{~kg} / \mathrm{cm}$ The equation of simple harmonic…

A string has mass per unit length of $10^{-6} \mathrm{~kg} / \mathrm{cm}$ The equation of simple harmonic wave produced in it is $Y=0.2 \sin (2 x+80 t) \mathrm{m}$. The tension in the string is
  1. 0.16 N
  2. 0.0016 N
  3. 1.6 N
  4. 16 N

Solution

Wave velocity, $v=\sqrt{\frac{T}{\mu}}$ $\mathrm{T}=\mathrm{v}^2 \mu=\mathrm{v}^2 \times 10^{-6}$
Given, $\mu=10^{-6} \mathrm{~kg} / \mathrm{cm}=10^{-4} \mathrm{~kg} / \mathrm{m}$ $\mathrm{T}=\mathrm{v}^2 \times 10^{-4}...(i)$
Given, $Y=0.2 \sin (2 x+80 t)$, Comparing with standard equation, $y=A \sin (K x+\omega t) \Rightarrow \omega=80$ $\begin{aligned} \therefore \quad \mathrm{K} & =\frac{\omega}{\mathrm{v}} \\ \mathrm{v} & =\frac{\omega}{\mathrm{K}} \end{aligned}$ $\begin{aligned} & v=\frac{80}{2} \\ & v=40 \mathrm{~m} / \mathrm{s} \end{aligned}$
From equation (i) $\begin{aligned} & \mathrm{T}=(40)^2 \times 10^{-4} \\ & \mathrm{~T}=0.16 \mathrm{~N} \end{aligned}$

Asked in: MHT CET 2024 (15 May Shift 2)

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