Question

The linear density of a vibrating string is 1.3×10⁴kg/m. A transverse wave is propagating on the string and is described by the equation y = 0.021 sin (30t + x), where x, y are measured in m and t in s. Find the tension in the wire

Answer 1

General equation of progressive wave:

$$\begin{gathered} y=A\sin \left(\omega t+kx\right) \\ \mathrm{Here},A\mathrm{is}\mathrm{amplitude} \\ \mathrm{\omega }\mathrm{is}\mathrm{angular}\mathrm{frequency} \\ \mathrm{\kappa }\mathrm{is}\mathrm{propagation}\mathrm{constant} \\ t\mathrm{is}\mathrm{time} \\ x\mathrm{is}\mathrm{displacement} \end{gathered}$$

Given equation:


$y=0.021\sin \left(30t+x\right)$

Compare with general equation:

$$\begin{gathered} A=0.021\mathrm{m} \\ \mathrm{\omega }=30\mathrm{rad}/\mathrm{s} \\ \mathrm{\kappa }=1{\mathrm{m}}^{-1} \end{gathered}$$

Velocity of the wave in the string:

$$\begin{gathered} v=\frac{\omega }{k} \\ =\frac{30}{1} \\ =30\mathrm{m}/\mathrm{s} \end{gathered}$$

Relation for velocity velocity in terms of tension:

$$\begin{gathered} v=\sqrt{\frac{T}{\mu }} \\ \mathrm{Here},T\mathrm{is}\mathrm{tension}\mathrm{and}\mu \mathrm{is}\mathrm{line}\mathrm{ar}\mathrm{density}. \end{gathered}$$

Expression for tension:

$$\begin{gathered} T=\mu v^2 \\ =\left(1.3\times {10}^{-4}\right){\left(30\right)}^2 \\ =0.12\mathrm{N} \end{gathered}$$