A125 cm length of string has mass 2.00 gand tension 7.00 N. (a) What is the wave speed for this string? (b)What is the lowest resonant frequency of this string?

The length of the string isL = 125 cm or 1.25 m.

The mass of the string is M = 2.00 g or 0.002 kg .

The tension in the string is $\tau =7.00\mathrm{N}$.

By using the formulas for the wave speed and the lowest resonant frequency, we can find the wave speed and the lowest resonant frequency of the string respectively.

Formula:

The speed of the wave $v=\sqrt{\frac{\tau }{\mu }}..........\left(1\right)$

The linear density of the string,$\mu =\frac{M}{L}.........\left(2\right)$

The resonant frequency of the wave, $f=\frac{v}{\lambda }.........\left(3\right)$

Using equation (2) in equation (1), we get the speed of the wave as given:

$$\begin{gathered} v=\sqrt{\frac{\tau L}{M}} \\ =\sqrt{\frac{7.00\times 1.25}{0.002}} \\ =66.1\mathrm{m}/\mathrm{s} \end{gathered}$$

Hence, the speed of the wave is 66.1 m/s

But, the wavelength of the wave with lowest resonant frequency$f_1$is${\lambda }_1=2L$, therefore, using this in the equation (3) and the given values, we get the lowest frequency as given:

$$\begin{gathered} f_1=\frac{v}{2L} \\ =\frac{66.1}{2\times 1.25} \\ =26.4\mathrm{Hz} \end{gathered}$$

Hence, the value of lowest frequency is 26.4 Hz