Question

A 1.50 mwire has a mass of 8.70 gand is under a tension of 120 N. The wire is held rigidly at both ends and set into oscillation. (a) What is the speed of waves on the wire? What is the wavelength of the waves that produce (b) one-loop and (c) two loop standing waves? What is the frequency of the waves that produce (d) one-loop and (e) two-loop standing waves?

Short answer

1. Speed of wave is 144 m/s
2. Wavelength that produced by one loop is 3.00 m
3. Wavelength that produced by two loop is 1.5 m
4. Frequency of one loop is 48.0 Hz
5. Frequency of two loop is 96 Hz

Step-by-step solution

Given data

Length of the wire, L=1.50 m

Mass of the wire, m=8.70 g or 0.00870 kg

Tension in the wire, T=120 N

Understanding the concept of the wave motion

Use the formula for speed in terms of tension and mass per unit length. To find the wavelength, we use the length of the wire. To find the frequency, use velocity from part (a) and wavelength from part (b) and (c).

Formula:

The speed of a string,$v=\sqrt{\frac{T}{\mu }}..........\left(1\right)$

The speed of a wave,$v=f\times \lambda ........\left(2\right)$

The linear density of a string,$\mu =\frac{m}{L}...........\left(3\right)$

Step 3(a): Calculation of speed of wave

Using equation (3), we get the linear density as follows:

$\begin{array}{l}\mu =\frac{0.0087}{1.5}\\ =5.8\times {10}^{-3}kg/\mathrm{m}\end{array}$

Now, using the given values in equation (1), we get the speed of the wave as:

$$\begin{gathered} v=\sqrt{\frac{120}{5.8\times {10}^{-3}}} \\ =143.83\frac{m}{s} \\ \approx 144m/s \end{gathered}$$

Hence, the value of the speed is 144 m/s

Step 4(b): Calculation of wavelength for one-loop wave

Now the wavelength for one loop standing wave is given as:

$\begin{array}{l}{\lambda }_1=L\times 2\\ =1.5\times 2\\ =3.00m\end{array}$

Hence, the value of wavelength for one-loop wave is 3.00 m

Step 5(c): Calculation of wavelength for two-loop wave

Now for two loop standing wave

$$\begin{gathered} {\lambda }_2=L \\ =1.5m \end{gathered}$$

Hence, the value of wavelength for two loops is 1.5 m

Step 6(d): Calculation of frequency of one-loop wave

Now, using equation (2) and given values, we get the frequency of one loop as:

$\begin{array}{l}144={\mathrm{f}}_1\times 3\\ {\mathrm{f}}_1=48.0\mathrm{Hz}\end{array}$

Hence, the value of frequency for one loop is 48.0 Hz

Step 7(e): Calculation of frequency of two-loop wave

Now, frequency of two loop using equation (2) and the given values is given as:

$\begin{array}{l}144=f_2\times 1.5\\ f_2=96\mathrm{Hz}\end{array}$

Hence, the value of frequency of two-loop wave is 96 Hz