Vibrating Strings EX-9964 Xplorer GLX Page 2 of 6

Vibrating Strings

EQUIPMENT

INCLUDED:

1 / String Vibrator / WA-9857
1 / Physics String / SE-8050
1 / GLX Power Amplifier / PS-2006
1 / Force Sensor / PS-2104
1 / C-clamp (small) / SE-7286
1 / Patch Cords / SE-9750
1 / Tape Measure / SE-8712A
NOT INCLUDED, BUT REQUIRED:
1 / Xplorer GLX / PS-2002
1 / DataStudio Software / CI-6870

INTRODUCTION

The general appearance of waves can be shown by means of standing waves in a string. This type of wave is very important because most of the vibrations of extended bodies, such as the prongs of a tuning fork or the strings of a piano, are standing waves. The purpose of this experiment is to study how the speed of the wave in a vibrating string is affected by the stretching force and the frequency of the wave.

THEORY

Standing waves (stationary waves) are produced by the interference of two traveling waves, both of which have the same wavelength, speed and amplitude, but travel in opposite directions through the same medium. The necessary conditions for the production of standing waves can be met in the case of a stretched string by having waves set up by some vibrating body, reflected at the end of the string and then interfering with the oncoming waves.

Standing Waves In Strings

A stretched string has many natural modes of vibration (three examples are shown above). If the string is fixed at both ends then there must be a node at each end. It may vibrate as a single segment, in which case the length (L) of the string is equal to 1/2 the wavelength (λ) of the wave. It may also vibrate in two segments with a node at each end and one node in the middle; then the wavelength is equal to the length of the string. It may also vibrate with a larger integer number of segments. In every case, the length of the string equals some integer number of half wavelengths. If you drive a stretched string at an arbitrary frequency, you will probably not see any particular mode; many modes will be mixed together. But, if the tension and the string's length are correctly

adjusted to the frequency of the driving vibrator, one vibrational mode will occur at a much greater amplitude than the other modes.

For any wave with wavelength λ and frequency f, the speed, v, is

v = λf (1)

The speed of a wave on a string is also given by

(2)

where F is the tension in the string and μ is the linear density (mass/length) of the string.

L is the length of the string and n is the number of segments. (Note that n is not the number of nodes). Since a segment is 1/2 wavelength then

(3)

Setting the wave speed in Equation (1) equal to the wave speed in Equation (2) and solving for the tension gives

(4)

Substituting for the wavelength from Equation (3) yields

(5)

In this experiment, the length and the number of segments will be held constant while the frequency is varied. The tension required to achieve 2 segments will be measured for various driving frequencies. Equation (5) shows that a graph of tension versus the square of the frequency will result in a straight line with

(6)
SETUP

In this experiment, standing waves are set up in a stretched string by the vibrations of an electrically-driven String Vibrator. The arrangement of the apparatus is shown below. The tension in the string is provided by a person pulling on the end of the string with a Force Sensor.

1. Measure the exact length of a piece of string several meters long. Measure the mass of the string and calculate the linear density, μ (mass/length). (If your balance is not precise enough to measure that length of string, use a much longer piece of string to calculate the linear density.)

2. As shown in the Figure 1, clamp the String Vibrator to the table. Attach about a meter of string to the vibrating blade and tie the other end of the string to the Force Sensor. Stretch the string taught and measure the length of the string.

Figure 1: Setup

3. Connect the GLX Power Amplifier to the String Vibrator. Plug the Power Amplifier into the 2 jacks on the side of the GLX.

4. Open the GLX file called "string vibrator".

PROCEDURE: FIXED STRING LENGTH

1. Open the Graph screen. Select Tools (F3) and Power Amp Config. This brings up the Power Amplifier settings below the graph. Press the Esc button to toggle between the graph and the Power Amplifier settings. Set the frequency of the Power Amplifier to 40 Hz and the amplitude to 6.0 V. Turn the Power Amplifier on.

2. Start recording on the GLX. Adjust the tension by pulling the string with the Force Sensor until the string vibrates in 2 segments. Adjust the tension further to achieve a “clean” node at the center. Also check the end of the vibrating blade; the point where the string attaches should be a node. It is more important to have a good node at the blade than it is to have the largest amplitude possible. However, it is desirable to have the largest amplitude possible while keeping a good node.

3. When the string is vibrating in exactly 2 segments, press the Flag button on the GLX to record a data point.

4. Increase the Power Amplifier frequency by 10 Hz. Then pull on the Force Sensor to once again achieve 2 segments in the string. Then press the Flag button to record a data point.

5. Repeat Step 4 until the frequency is 130 Hz. Then press the Start/Stop button and close the Power Amplifier Configuration window by choosing Tools (F3) and toggling the Power Amp Config.

ANALYSIS

1. On the Graph screen, select Tools (F3) and Linear Fit (5). Record the slope of the line.

2. Use Equation (6) to calculate the linear density of the string. Compare this value with the value measured directly with a scale. Calculate the percent difference.

FURTHER INVESTIGATIONS

1. For a fixed frequency, vary the tension to achieve different numbers of segments.

2. For a fixed tension, vary the frequency to achieve different numbers of segments.

3. Repeat the experiment using a different density string.


QUESTIONS

1. Calculate the wavelength of the wave in the string.

2. Calculate the speed of the wave in the string for 40 Hz and two segments. Then calculate the speed of the wave in the string for 120 Hz and two segments.

(a) Why are their two ways of calculating the speed?

(b) By what percentage does the speed change when the frequency is increased by a factor of 3 (from 40 Hz to 120 Hz)? Does the speed increase or decrease?

APPENDIX: XPLORER GLX CONFIGURATION

1. Plug the Force Sensor into the GLX.

2. Plug the GLX Power Amplifier into the GLX.

3. On the main menu, go to Sensor (F4) and change the Sample Rate to 500 Hz and the Smooth Averaging to 50 Points. Choose the Force to be Not Visible and the Tension to be Visible. Change the Mode (F1) to Manual.

4. Activate the GLX Power Amplifier: From the home screen, go to the Output screen and change the Output Device from Internal Speaker to Power Amplifier. The GLX will then quickly calibrate the Power Amplifier.

5. Set the Waveform to a Sine wave with Amplitude equal to 6.00V, and Frequency equal to 40.0 Hz.

6. From the main menu, go to the Calculator (F3) and make the following calculation:

freq squared = [Output Frequency (Hz)]^2

On the Calculator screen, choose Edit (F4) and Data Properties (6). Edit the Units to read Hz^2. Then press OK (F1) to accept the changes.

7. From the main menu, select Graph (F1) and make a graph of Tension vs. freq squared.

8. Open the Data Files window on the main menu and name the file "string vibrator" and save it. Return to the Graph window. The GLX is now ready to record data.

Written by Ann Hanks