Question

A piano wire weighing 6⋅00 g and having a length of 90⋅0 cm emits a fundamental frequency corresponding to the "Middle C" $\left(\nu =261·63\mathrm{Hz}\right)$. Find the tension in the wire.

Answer 1

Given:
Length of the piano wire (L)= 90.0 cm = 0.90 m
Mass of the wire = 6.00 g = 0.006 kg
Fundamental frequency (f_o) = 261.63 Hz

$$\begin{gathered} \mathrm{Linear}\mathrm{mass}\mathrm{density},m=\left(\frac{6}{90}\right)\mathrm{gm}/\mathrm{cm} \\ =\left\{\frac{6\times {10}^{-3}}{90\times {10}^{-2}}\right\}\mathrm{kg}/\mathrm{m} \\ =\left(\frac{6}{900}\right)\mathrm{kg}/\mathrm{m} \end{gathered}$$

$\mathrm{Fundamental}\mathrm{f}\mathrm{requency},f_o=\frac{1}{2L}\sqrt{\left(\frac{T}{m}\right)}$
$$\begin{gathered} \Rightarrow 261.63=\frac{1}{\left(2\times 0.09\right)}\sqrt{\left\{\frac{T\times 900}{6}\right\}} \\ \Rightarrow 0.18\times 261.63=\sqrt{150T} \\ \Rightarrow 150\mathit{}T={\left(261.63\times 0.18\right)}^2 \\ \Rightarrow T=\frac{{\left(261.63\times 0.18\right)}^2}{150} \\ =1478.52\mathrm{N}\approx 1480\mathrm{N} \end{gathered}$$

Hence, the tension in the piano wire is 1480 N.