Question

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.

Short answer

1. The speed of the waves on the string is 82 m/s
2. The longest possible wavelength for a standing wave is 16.8 m
3. The frequency of that wave is 4.88Hz

Step-by-step solution

Given data

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

Understanding the concept of wave motion

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

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

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

Step 4(b): Calculation of longest possible wavelength

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

Step 5(c): Calculation of frequency

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