In this laboratory students will observe standing transverse waves on a string. They will determine the wave velocity v=fl from the frequency and wavelength of each wave. Given the string tension F they will calculate the linear density (mass per unit length) of the string.
A standing wave is a pattern which results from the interference of two or more waves traveling in the same medium. All standing waves are characterized by positions along the medium which are standing still. Such positions are referred to as nodes. Standing waves are also characterized by antinodes. These are positions along the medium where the particles oscillate about their equilibrium position with maximum amplitude. Standing wave patterns are always characterized by an alternating pattern of nodes and antinodes.
Standing waves of many different wavelengths can be produced. Each wavelength corresponds to a particular frequency and is known as a harmonic. Wavelength and frequency are related through lf=v, where v is the speed of wave. The shorter the wavelength, the higher is the frequency. The lowest possible frequency of a standing wave is known as the fundamental frequency or the first harmonic. The second lowest frequency is known as the second harmonic, the third lowest frequency is known as the third harmonic, and so on.
Standing waves of many different wavelengths can be produced on a string with two fixed ends, as long as an integral number of half wavelength fits into the length of the string. For a standing wave on a string of length L with two fixed ends
L=n(l/2), n=1,2,3,... .
 | Fundamental: L=l/2, n=1, 1/2 wavelength fits into the length of the string. |
| Second harmonic: L=l n=2. one wavelength fits into the length of the string. |
| Third harmonic: L=3l/2, n=3, 3/2 wavelengths fit into the length of the string. |
For a string, the speed of the waves is a function of the mass per unit length m=m/L of the string and the tension F in the string
In this lab waves on a string with two fixed ends will be generated by a string vibrator. The waves will have a frequency of 120 Hz. Their wavelength is given by l=v/f. Students will analyze video clips in which the string tension F is fixed and the length of the string is being varied. Students will measure the length of the string when the string supports a standing wave such that 1, 2, or 3 half wavelength of a wave fit into the length of the string. Then 120 Hz is a natural frequency of the string and the vibrator drives the string into resonance. The amplitude increases and the standing waves can easily be observed.
You will analyze three video clips, string_x.avi, x-1-3. To play a video clip or to step through it frame-by-frame click the "Begin" button. The "Video Analysis" web page will open. You can toggle between the current page and the "Video Analysis" page by pressing Alt-Tab.
In the video clips the vibrator is mounted onto a rod which is fixed to the table with a clamp. A pulley is mounted onto another rod on a movable stand. One end of a string is attached to the vibrator. The string is passed from the vibrator over the pulley and a mass is attached to its other end. The string is level.
The string is a string with two fixed ends. The amplitude of the vibrator arm is so small compared to the amplitude of the string at resonance, that the vibrator is very close to a node. The other node is the top of the pulley.
| Step through the video clip string_1.avi frame by frame. You will see the vibrator drive the string into resonance when the length of the string becomes equal to exactly 1/2 wavelength of a standing wave. Locate the frame in which this happens. | ||||||||||||||||||||||||||||
| Choose to measure the x-coordinate of two objects and click to start taking data. Click to enter the x-coordinate of the node at the vibrator and the node on top of the pulley into the spreadsheet that has opened. Then stop taking data and calibrate your data. Find the distance between the two nodes L. | ||||||||||||||||||||||||||||
| Open Excel and construct a spreadsheet as shown below. Enter
L, l and the string tension F given in the
clip. For the fundamental l=2L.
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| Repeat the experiment using the video clips string_2.avi and string_3.avi. These clips show the second and third harmonic, respectively. For the second harmonic l=L and for the third harmonic l=2L/3. | ||||||||||||||||||||||||||||
| For waves on a string we have F=mv2. This tension is provided by a hanging mass, F=mg. Calculate v and v2 and plot F versus v2. Use Excel's regression function to find the slope of the best fitting straight line to this plot. This slope is equal to the mass per unit length of the string m. The uncertainty in the slope equals the uncertainty in m. |
Open Microsoft Word and prepare a report using the template shown below.
| In a few sentences summarize the experiment. |
| Show your spreadsheet and your plot. |
| What is the mass per unit length m of the string in units of kg/m and what is the uncertainty in this value as determined from your measurements. |
| Refer to the video clip string_1.avi. How much hanging mass would be needed to produce a resonance of the fundamental standing wave, if the string had same length L but only half the linear density m? |
| Refer to the video clip string_1.avi. How much hanging mass would be needed to produce a resonance of the fundamental standing wave, if the string had same linear density m but twice the length L? |
Save your Word document (your name_lab10.doc) and attach it to an e-mail message to mbreinig@utk.edu.