Determining the Speed of Sound in a String - Lab Report Example

Running head: To determine the speed of sound in a string Student’s name Institution Course Professor Date of Experiment Date of Submission Abstract This report describes the experimental procedures that were used to determine the velocity of sound in a string by investigating standing waves in a string. Equipment such as a vibrating device, a string, a set of weights, and a tension measuring device, a tape measure and a top loading balance were familiarized with. The oscillator was set to a frequency of 15Hz and the string was tension with a 200g weight. The frequency was adjusted till a standing wave was produced. Graphs of frequency and inverse of wavelength were plotted and a slope was calculated and gave the value of the speed of sound for each tension (C). Both tabulated and calculated values of the speed of sound in string were compared. Table of Contents Abstract 2 Table of Contents 3 1.0Introduction 4 1.1Theory 4 2.0Methodology 6 3.0Procedures 6 4.0Results and calculations 7 5.0Conclusions 13 5.1Sources of errors 14 6.0References 15 1.0Introduction This report describes the experimental procedures that were used to determine the velocity of sound in a string by investigating standing waves in a string. The relationship between the tension in the string and the wavelength in the standing wave was investigated. Furthermore, the relationship between the frequency of oscillation of the string and the number of oscillation in the wave was carefully studied. The oscillator was set to a frequency of 15Hz and the string was tension with a 200g weight. The frequency was adjusted till a standing wave was produced. These relationships could help to find the linear mass density of the string (Lerner, 1996). The aims of the experiment were to: 1) Familiarize with the following equipment: a vibrating device, a string, a set of weights, and a tension measuring device, a tape measure and a top loading balance. 2) Develop the research skills 3) Develop an understanding of the standing waves 4) Investigate the speed of sound in a string. 1.1Theory A standing wave is formed when a boundary blocks further propagation of the wave which leads to a wave reflection which introduces a counter-propagating wave (Lerner, 1996). The wavelength of can be computed by setting up standing wave of a known length in the string (Tohyama, 2011). Plucking a string under tension will result to production of transverse waves. In a given harmonic, wavelength can be expressed as λ= 2L (1/n) Where L = Length of the stretched string n= Number of oscillation in the string Linear mass density of the string can be achieved by measuring the mass and length of the string. Moreover, it can be found by inquiring the relationship between the tension, frequency, length of the string and the wavelength in the standing wave (Lerner, 1996). The velocity of the wave in a string can be expressed as C=f λ Where C= The speed of sound in ms1 f= The frequency in Hz λ= The wavelength in m In a stretched string the speed of a wave becomes C = 2 L f (1/n) Where f= frequency of the wave, n=number of segments in a string Alternatively, the speed of a wave travelling in a string depends on the linear mass density, µ of the string and the tension, T in the string. C=√ (T/µ) C= The speed of sound in the string in m/s1 T=The tension in the string in N µ= The mass per unit length (linear density) of the string in kg/m1 2.0Methodology Materials and equipment employed during the experiment were a) A vibrating device b) A string c) A set of weights d) A tension measuring device e) A tape measure f) A top loading balance 3.0Procedures First of all, the oscillator was set to a frequency of 15Hz and the string was tensioned with a 200g weight. The frequency of the string was adjusted till a standing wave was produced. The end of the vibrating blade was checked to ensure that a node was present at the point where the string attaches. The blade rattling against the case indicated a bad node. The wavelength of the standing wave was carefully measured. The frequency of the string was changed in order to find five different standing waves. Each time, the weight used, the tension in the string, the frequency of the oscillator and the wavelength of the standing wave were recorded. This experiment was repeated with other five different weights. The length of the string was measured and a value for its mass per unit length in kg/m1. For each tension, a graph of frequency against inverse of wavelength was plotted in order to obtain the speed of sound in m/s1. 4.0Results and calculations The results from the experiments were tabulated as shown in the tables below a. 200g Frequency (Hz) Tension (N) Wavelength, λ (m) 1/λ 15 2.2 2.4(2/3) = 1.6 0.625 20 2.2 2.4(2/4) =1.2 0.83 25 2.2 2.4(2/5) =0.96 1.042 30 2.2 2.4(2/6) =0.8 1.25 35 2.2 2.4(2/7) =0.69 1.45 b. 100g Frequency (Hz) Tension (N) Wavelength, λ (m) 1/λ 15 1.3 2.4(2/4) = 1.2 0.83 19 1.3 2.4(2/5) = 0.96 1.042 23 1.3 2.4(2/6) =0.8 1.25 27 1.3 2.4(2/7) = 0.69 1.45 31 1.3 2.4(2/8) = 0.6 1.667 c. 150g Frequency (Hz) Tension (N) Wavelength, λ (m) 1/λ 13.1 1.8 2.4(2/3) = 1.6 0.625 15.1 1.8 2.4(2/7) = 0.69 1.45 17.2 1.8 2.4(2/4) = 1.2 0.83 22 1.8 2.4(2/5) = 0.96 1.042 26.2 1.8 2.4(2/6) = 0.8 1.25 d. 250g Frequency (Hz) Tension (N) Wavelength, λ (m) 1/λ 17 2.8 2.4(2/3) =1.6 0.625 22.3 2.8 2.4(2/4) =1.2 0.83 28 2.8 2.4(2/5) =0.96 1.042 39 2.8 2.4(2/6) =0.8 1.25 39.6 2.8 2.4(2/7) =0.69 1.45 In part one The length of the string= 2.4m The weight of the string = 13g The linear mass density = 4.44 * 10-3 kg/m In a mass of 200g the gradient was solved by = (35.68-15.64)/ (1.453-0.626) 20.04/0.727 27.57 In a mass of 100g the gradient was solved by = (31.31-15.54)/ (1.673-0.830) 15.77/0.830 18.71 In a mass of 150g the gradient was solved by = (26.68-13.41)/ (1.254-0.626) 13.27/0.628 21.13 In a mass of 250g the gradient was solved by = (34.42-17.59)/ (1.193-0.627) 16.93/0.566 29.73 In part two, the speed of sound on the string for each tension can be calculated as C=√ (T/µ) C= The speed of sound in the string in m/s1 T=The tension in the string in N µ= The mass per unit length (linear density) of the string in kg/m1 In 200g, Tension=2.2N and linear density= 4.44 * 10-3 So C= (2200/4.44)1/2 22.26m/s In 100g, Tension=1.3N and linear density= 4.44 * 10-3 So C= (1300/4.44)1/2 17.11m/s In 150g, Tension=1.8N and linear density= 4.44 * 10-3 So C= (1800/4.44)1/2 20.13m/s In 250g, Tension=2.8N and linear density= 4.44 * 10-3 So C= (2800/4.44)1/2 25.11m/s Comparison of the values of C (speed of sound in a string) Mass in the string Value of C in part 1 Value of C in part 2 Percentage error (%) 200g 27.57 22.26 19.26% 100g 18.71 17.11 8.55% 150g 21.13 20.13 4.73% 250g 29.73 25.11 15.5% From the above table, the percentage error for the values of the speed of sound in the string is negligible. The speed of a travelling wave in a stretched string can be calculated by the mass per unit length of the string and the tension of the string. A standing wave is form when two equal waves travel in opposite directions along a line. 5.0Conclusions The speed of a wave traveling along a vibrating string depends on the linear density and the tension of the string. The greater the tension placed on the string, the faster the waves can travel down the length of the string. There is relationship between the tension in the string and the wavelength in the standing wave and also, the relationship between the frequency of oscillation of the string and the number of segments in the wave. 5.1Sources of errors The sources of errors in the experiment can be attributed to the weighing error, the linear density of the string is non-uniform, parallax (reading the position of nodes) and the frequency of vibration is not constant. 6.0References Lerner,” Physics: for scientists and engineers; Modern Physics: for scientists and engineers”, ones & Bartlett Learning, 1996.print. Tohyama, “Sounds and Signals: signals and communication Technology”, Springer, 2011 .print Vern J. Ostdiek, Donald J. Bord,”Inquiry Into Physics”, Cengage Learning, 2012.print Read More