The Speed of Sound on a String

The frequency of a plucked string depends on its mass per unit length, its tension and its length. These variables allow for the multitude of pitches produced by stringed instruments. We will determine how the speed of sound on a string depends on those variables by varying them and measuring the resulting frequency of the fundamental mode, using the equation
ν = c / 2 L
to determine the speed of sound from the closed-closed boundary conditions. We will use a sonometer with a detector coil, a 1 kg mass, a level and an oscilloscope. The strings used on the sonometer have linear mass densities of
diameter in inchesμ in gm/cm
.010.0039
.014.0078
.017.0112
.020.0150
.022.0184

Name:

Lab Partners:

When entering numeric data, use exponentials: ie., 1.6 * 10-19 = 1.6E-19.

Procedure

  1. Install the .010" diameter string in the sonometer. Place a 1 kg mass on the 1st slot on the tension lever (use a paper clip to make sure the mass clears the sonometer). Place the bridges under the string so that the distance between them is exactly 60 cm. Adjust the string tension until the lever is exactly horizontal (a mass of M on the nth slot produces a tension of n M g when the lever is horizontal).

    You will have to re-level the tension lever each time a parameter is changed.

  2. Place the detector coil exactly midway between the bridges. With the detector coil plugged into the oscilloscope and the volts/div set to maximum sensitivity, lightly pluck the string just above the detector coil. Adjust the time/div until the display shows a between one and two full waves, and record the time/div setting and the horizontal length of a single wave in divisions:
    t/div1 = ms/div, ldiv1 = divisions

  3. Repeat with the 1 kg mass on the second through fifth slots, being sure to re-level the tension lever each time:
    t/div2 = ms/div, ldiv2 = divisions

    t/div3 = ms/div, ldiv3 = divisions

    t/div4 = ms/div, ldiv4 = divisions

    t/div5 = ms/div, ldiv5 = divisions

  4. Place the 1 kg mass on the first slot again and repeat the experiment, this time varying the length of string between the bridges at 40, 45, 50 and 55 cm. Be sure the detector coil is exactly midway between the bridges at each trial.
    t/div40 = ms/div, ldiv40 = divisions

    t/div45 = ms/div, ldiv45 = divisions

    t/div50 = ms/div, ldiv50 = divisions

    t/div55 = ms/div, ldiv55 = divisions

  5. Finally, repeat the experiment for each of the other strings, with the 1 kg mass on the first slot (leveling the lever each time), and the distance between the bridges fixed at 60 cm (the subscripts below are the string diameters in thousandths of an inch):
    t/div14 = ms/div, ldiv14 = divisions

    t/div17 = ms/div, ldiv17 = divisions

    t/div20 = ms/div, ldiv20 = divisions

    t/div22 = ms/div, ldiv22 = divisions

Analysis

  1. Compute the speed of sound (in cm/s) as a function of tension (in N) from the first set of trials using
    ci = 2 * 60 / (t/divi * ldivi / 1000)
    and
    Ti = i * 980,000

    c1 = cm/s, T1 = dynes

    c2 = cm/s, T2 = dynes

    c3 = cm/s, T3 = dynes

    c4 = cm/s, T4 = dynes

    c5 = cm/s, T5 = dynes

    What are the units on the numerical factor of 1000?

  2. Compute the speed of sound (in cm/s) as a function of length (in cm) from the second set of trials using
    ci = 2 * i / (t/divi * ldivi / 1000)

    c40 = cm/s

    c45 = cm/s

    c50 = cm/s

    c55 = cm/s

    c60 = cm/s

    (c60 is the same as c1 from the first trials)

  3. Compute the speed of sound (in cm/s) as a function of linear mass density (in gm/m) from the third set of trials using
    ci = 2 * 60 / (t/divi * ldivi / 1000)

    c10 = cm/s

    c14 = cm/s

    c17 = cm/s

    c20 = cm/s

    c22 = cm/s

    (c10 is the same as c1 from the first trials)

  4. Plot the data (Ti, ci) from the first set of trials. How does the speed of sound depend on tension?

  5. Plot the data (li, ci) from the second set of trials. How does the speed of sound depend on length?

  6. Plot the data (μi, ci) (using the table at the beginning) from the third set of trials. How does the speed of sound depend on linear mass density?

  7. What is the equation for speed of sound as a function of one or more of the variables we have measured?

©2005, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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