Question

A transverse traveling wave on a taut wire has an amplitude of0.200 mm and a frequency of 530 Hz. Ittravels with a speed of 196 m/s.

(a) If the wave equation is written in the formy = A sin(kx - ωt), whatare the parameters A, k, andω?
1 m
2 rad/m
3 rad/s

(b) The mass per unit length of this wire is 3.50 g/m. Find the tension in the wire.
4 N

Answer 1

$$ \begin{array}{l} \begin{array}{l} \text { Amplitude }(A)=0.2 \mathrm{~mm} \\ =2 \times 10^{-4} \mathrm{~m} \end{array} \\ \text { Frequency }(f)=530 \mathrm{~Hz} \\ \text { Speed }(v)=196 \mathrm{~m} / \mathrm{s} \end{array} $$

Angular Frequency \(\omega=2 \pi f\)

$$ \begin{array}{l} =2 \pi(530 \mathrm{~Hz}) \\ =3330 \mathrm{rad} / \mathrm{s} \end{array} $$

Angular wave number \(k=\frac{\omega}{v}\)

$$ \begin{array}{l} =\left(\frac{3330 \mathrm{rad} / \mathrm{s}}{196 \mathrm{~m} / \mathrm{s}}\right) \\ =16.9 \mathrm{rad} \mathrm{m} \end{array} $$

(b) \(\quad\) Mass per unit leng th of the wire \((\mu)=3.5 \mathrm{~g} / \mathrm{m}\)

$$ =3.5 \times 10^{-3} \mathrm{~kg} / \mathrm{m} $$

Tension in the string \(T=v^{2} \mu\)

$$ \begin{array}{l} =(196 \mathrm{~m} / \mathrm{s})^{2}\left(3.5 \times 10^{-3} \mathrm{~kg} / \mathrm{m}\right) \\ =134.4 \mathrm{~N} \end{array} $$