Physics - Wave on String Question with Solution | TestHub

Given:
Linear mass density, $\mu =0.1{\text{ kg m}}^{-1}$
Length of the string, $L=0.9\text{ m}$
Tension, $T=40\text{ N}$
Number of segments, $n=3$
Amplitude, $A=0.3\text{ cm}=3\times 10^{-3}\text{ m}$

The velocity of the transverse wave on the string is:
$v=\sqrt{\frac{T}{\mu }}=\sqrt{\frac{40}{0.1}}=\sqrt{400}=20\text{ m/s}$

Since the string vibrates in $3$ segments, the frequency of vibration is:
$f=\frac{nv}{2L}=\frac{3\times 20}{2\times 0.9}=\frac{60}{1.8}=\frac{100}{3}\text{ Hz}$

The angular frequency is:
$\omega =2\pi f=2\pi \times \frac{100}{3}=\frac{200\pi }{3}\text{ rad/s}$

The maximum particle velocity is given by:
$v_{\text{max}}=A\omega$

Substituting the values:
$v_{\text{max}}=(3\times 10^{-3})\times \left(\frac{200\pi }{3}\right)$

$v_{\text{max}}=10^{-3}\times 200\pi =0.2\pi =\frac{\pi }{5}\text{ m/s}$

Answer: $\frac{\pi }{5}$