A string fixed at both ends is 8.40 mlong and has a mass of 0.120 kg. It is subjected to a tension of 96.0 Nand set oscillating. (a) What is the speed of the waves on the string? (b) What is the longest possible wavelength for a standing wave? (c) Give the frequency of that wave.

The length of the string is L = 8.40 m

The mass of the string is M = 0.120 kg

The tension in the string is T = 96.0 N

By finding the wave speed and the wavelength from the linear mass density and the length of the string respectively, we can find the frequency of the wave.

Formula:

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

The frequency of the wave,$\mu =\frac{M}{L}.....\left(2\right)$

The linear density of the string, $f=\frac{v}{\lambda }.....\left(3\right)$

Using equation (1) and (2), the speed of the wave is given as:

$$\begin{gathered} v=\sqrt{\frac{\tau L}{M}} \\ =\sqrt{\frac{96.0\times 8.40}{0.120}} \\ =82m/s \end{gathered}$$

Hence, the speed of the wave is 82 m/s

The longest possible wavelength $\lambda$for a standing wave is related to the length of the string by

$L=\frac{\lambda }{2}$

So, the wavelength $\lambda$is given by:

$$\begin{gathered} \lambda =2L \\ =2\times 8.40 \\ =16.8m \end{gathered}$$

Hence, the value of the longest wavelength is 16.8 m

The frequency f using equation (3) and the given values is given by:

$$\begin{gathered} f=\frac{v}{\lambda } \\ f=\frac{82.0}{16.8} \\ =4.88\mathrm{Hz} \end{gathered}$$

Hence, the value of the frequency of that wave is 4.88 Hz