Question

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?

Short answer

1. The wave speed for this string is 66.1 m/s
2. The lowest frequency of this string is 26.4 Hz

Step-by-step solution

Given data

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 $\tau =7.00\mathrm{N}$.

Understanding the concept of resonant frequency

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, $f=\frac{v}{\lambda }.........\left(3\right)$

Step 3(a): Calculation of the speed of the wave

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

Step 4(b): Calculation of the lowest frequency

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