Question

A one-metre long stretched string having a mass of 40 g is attached to a tuning fork. The fork vibrates at 128 Hz in a direction perpendicular to the string. What should be the tension in the string if it is to vibrate in four loops?

Answer 1

Given:
Length of the stretched string (L) = 1.00 m
Mass of the string =40 g
String is attached to the tuning fork that vibrates at the frequency (f) = 128 Hz
Linear mass density (m) $=\left(40\times {10}^{-3}\right)\mathrm{kg}/\mathrm{m}$
No. of loops formed, (n) = 4
$$\begin{gathered} L=\frac{n\lambda }{2} \\ \Rightarrow \lambda =\frac{2L}{n}=\frac{2\times 1}{4} \\ \Rightarrow \lambda =0.5\mathrm{m} \\ \text{W}\mathrm{ave}\mathrm{speed}\left(v\right)=f\lambda =128\times 0.5 \\ \Rightarrow v=64\mathrm{m}/\mathrm{s} \end{gathered}$$

$$\begin{gathered} \mathrm{We}\mathrm{know}: \\ v=\sqrt{\left(\frac{T}{m}\right)} \\ \Rightarrow T={\nu }^{2}m \\ ={\left(64\right)}^2\times 40\times {10}^{-3} \\ =163.84\approx 164\mathrm{N} \end{gathered}$$
Hence, the tension in the string if it is to vibrate in four loops is 164 N.