Question 14.8: HARMONICS OF A STRETCHED WIRE GOAL Calculate string harmonic...

(a) Find the first three harmonics at the given tension.

Use Equation $13.18$

$v=\sqrt{\frac{F}{\mu}}$      [13.18]

to calculate the speed of the wave on the wire:

$v=\sqrt{\frac{F}{\mu}}=\sqrt{\frac{80.0  \mathrm{N}}{2.00 \times 10^{-3} \mathrm{~kg} / \mathrm{m}}}=2.00 \times 10^{2} \mathrm{~m} / \mathrm{s}$

Find the wire’s fundamental frequency from Equation 14.15:

$\begin{aligned}&f_{1}=\frac{v}{2 L}=\frac{2.00 \times 10^{2} \mathrm{~m} / \mathrm{s}}{2(1.00 \mathrm{~m})}=1.00 \times 10^{2} \mathrm{~Hz}\end{aligned}$

Find the next two harmonics by multiplication:

$\begin{aligned}&f_{2}=2 f_{1}=2.00 \times 10^{2} \mathrm{~Hz}, f_{3}=3 f_{1}=3.00 \times 10^{2} \mathrm{~Hz}\end{aligned}$

(b) Find the wavelength of the sound waves produced.

Solve $v_{s}=f \lambda$ for the wavelength and substitute the frequencies:

$\begin{aligned}&\lambda_{1}=v_{s} / f_{1}=(345 \mathrm{~m} / \mathrm{s}) /\left(1.00 \times 10^{2} \mathrm{~Hz}\right)=3.45 \mathrm{~m} \\&\lambda_{2}=v_{s} / f_{2}=(345 \mathrm{~m} / \mathrm{s}) /\left(2.00 \times 10^{2} \mathrm{~Hz}\right)=1.73 \mathrm{~m} \\&\lambda_{3}=v_{s} / f_{3}=(345 \mathrm{~m} / \mathrm{s}) /\left(3.00 \times 10^{2} \mathrm{~Hz}\right)=1.15 \mathrm{~m}\end{aligned}$

(c) Find the fundamental frequency corresponding to the elastic limit.

Calculate the tension in the wire from the elastic limit:

$\begin{aligned}&\frac{F}{A}=\text { elastic limit } \quad \rightarrow \quad F=(\text { elastic limit }) A \\&F=\left(2.80 \times 10^{8} \mathrm{~Pa}\right)\left(2.56 \times 10^{-7} \mathrm{~m}^{2}\right)=71.7 \mathrm{~N}\end{aligned}$

Substitute the values of $F, \mu$, and $L$ into Equation 14.16:

$\begin{aligned}&f_{1}=\frac{1}{2 L} \sqrt{\frac{F}{\mu}} \\&f_{1}=\frac{1}{2(1.00 \mathrm{~m})} \sqrt{\frac{71.7 \mathrm{~N}}{2.00 \times 10^{-3} \mathrm{~kg} / \mathrm{m}}}=94.7 \mathrm{~Hz}\end{aligned}$

REMARKS From the answer to part (c), it appears we need to choose a thicker wire or use a better grade of steel with a higher elastic limit. The frequency corresponding to the elastic limit is smaller than the fundamental!