Lab5 Standing Waves

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Toronto Metropolitan University *

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CPCS 130

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Physics

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Dec 6, 2023

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Standing Waves Objective : This experiment aims to investigate longitudinal and transverse standing waves in depth. Our specific objectives are to measure the linear mass density of a stretched string, compute the speed of standing transverse waves in a rope, and compute the standing longitudinal wave speed in a spring. We also aim to measure the speed of sound precisely and explore standing waves in a tube. This diverse approach will add to a more thorough understanding of wave phenomena by offering insightful information about the fundamentals and traits of standing waves in a variety of forms. Equipment: To guarantee that the experiments were carried out precisely and accurately, the following apparatus' are used: -Mechanical Wave Driver -Sine Wave Signal Generator -Standing Speed of Sound Apparatus -Function Generator -Measuring Tape, String, Spring & Weights -Oscilloscope 6.2 – Standing Waves in a Rope and a Spring Data: 6.2.1)-4 Standing Waves in a Rope Page 1 of 8

Mean: 33.11 m/s, Standard Deviation: 1.285 m/s Therefore, 31.83m/s & 34.29m/s 6.2.3)-1 Longitudinal Standing Waves in a Spring Mean: 2.552 m/s, Standard Deviation: 0.186 m/s Therefore, 2.366 m/s & 2.738 m/s Page 2 of 8

Data Analysis: 6.2.1) Actual Unstretched (L u ): (0.508 ± 0.005) m Stretched String (L s ): (0.84 ± 0.005) m Linear mass density: µ u = 0.0011 ± 0.0001 kg/m 2. m string =µ u L u = (0.0011 ± 0.0001) kg/m * (0.508 ± 0.005) m = 5.59x10 -4 kg δm string = 0.0001 kg / m 0.0011 kg / m + 0.005 m 0.508 m ¿ )(5.59x10 -4 kg) =5.63x10 -5 kg Therefore, (5.59x10 -4 ± 5.63x10 -5 ) kg 3. µ s = m string /L s = (5.59x10 -4 ± 5.63x10 -5 ) kg / (0.84 ± 0.005)m =6.65x10 -4 kg/m δ µ s = 5.63E-5 kg 5.59E-4 kg + 0.005 m 0.84 m ¿ )(6.6x10 -4 kg/m) = 7.04x10 -5 kg/m Therefore, (6.65x10 -4 ± 7.04x10 -5 ) kg/m 4. v = √ T / µs , T= (m hanging )(g)= 0.05kg x 9.81 m/s 2 =0.4905 N = √ ( 0.4905 ± 0.0005 N )( 6.65 E − 4 ± 7.04 E − 5 ) kg / m = √ 737.59 =27.16 m/s Page 3 of 8

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δv / v =( 1 / 2 )( ( δT T ) + ( δµs / µs ) ) δv = 1 2 ( 0.0005 N 0.4905 N ) +( 7.04 x 10 − 5 kg / m 6.65 x 10 − 4 kg / m ) (27.16 m/s) = 1 2 ( ( 0.001019 ) + ( 0.10586 ) ) (27.16 m/s) = 1.451 m/s Therefore, v= (27.16 ± 1.451) m/s 6.2.3) 1. v = √ k m spring L = ¿ √ 1.9 N m 0.0274 kg ¿ 0.33m) = 2.75 m/s Discussion & Answers to Questions: 6.2.2) 1. The measured wave speed range of the transverse standing waves is 31.83 m/s to 34.29 m/s. The calculated wave speed range of 25.71 m/s to 28.61 m/s proves that it is not within the uncertainty provided by the statistics fit in logger pro. Nevertheless, the values are closely aligned, indicating a discernible correlation in the numerical data. 2. If the tension and linear mass density of both ropes, no matter the size, are the same, the wavelength and frequency will vary but the wave velocity will stay the same. The wavelength and string length for a second harmonic are inversely related. Consequently, in comparison to the shorter rope, the longer rope will have a greater wavelength for the second harmonic. In addition, the frequency has an inverse relationship with string length and a direct relationship with wave velocity. The longer rope will have a lower frequency for the second harmonic compared to the shorter rope. Lastly, for Page 4 of 8

a particular tension and linear mass density, the wave velocity is independent of the string's length. Thus, assuming the same tension and linear mass density for the long and short ropes, the wave velocity will be the same. 3. In this scenario, observations can be made that an increase in frequency corresponds to a decrease in amplitude. One possible explanation for this observation lies within the mass hanging as gravity is pushing downward, causing strain in the rope. This tension may affect the systems damping properties. Tension is capable of increasing dissipation of energy at higher frequencies, resulting in a reduction of amplitude, since the energy is more efficiently expended. 6.2.4) 1. Percentage Difference = 2.75 − 2.552 2.552 ∗ 100 = 7.76% Hence, the wave speed I calculated closely aligns with the measured wave speed of the longitudinal standing waves. 2. The measured wave speed range of the longitudinal standing waves is 2.366 m/s to 2.738 m/s. The calculated wave speed range of 2.75 m/s shows that it is not within the uncertainty provided by the statistics fit in logger pro. However, the calculated value is off by 0.012 m/s are closely aligned, indicating a discernible correlation in the numerical data. 3. If the two springs are of the same length and the same linear mass density, but one is soft, and the other is stiff, there will be changes to two variables out of the three. Since the harmonic number and the spring length are the same for both springs, the wavelengths will be the same. These two elements are what essentially influence the wavelength. However, the increased rigidity of the stiffer spring is expected to facilitate faster wave transmission and induce greater tension. Consequently, the frequency and wave velocity are anticipated to increase in the stiffer spring compared to the softer counterpart. Page 5 of 8

6.3 – Sound standing waves in the air tube Data: 6.3.2) 1 to 6 Data Analysis: 6.3.2)-7 The measured value obtained through experimentation as a mean number is (344 ± 5.537) m/s giving a range of 338 m/s to 349 m/s. The standard value of 343 m/s for the speed of sound is within the range, therefore, confirming that there is accurate correlation between the experiment and standard value. Page 6 of 8

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Discussion & Answers to Questions: 6.3.3) 1. The speed of a sound standing wave in an air tube remains constant because any change in frequency is accompanied by a corresponding adjustment in wavelength, preserving the fundamental relationship between frequency, wavelength, and the constant speed of sound in the medium which is air. 2. The fundamental frequency is set by the constant piston position, and when you vary the incoming sound wave's frequency, you can experiment with various harmonics while preserving resonance conditions. Resonance can occur at different frequencies because each harmonic modifies its wavelength to fit within the constant length of the air column. With the wavelength and frequency numbers collected, you can then calculate a velocity to obtain approximately 343 m/s. 3. At a frequency of 700 Hz: -Here you can see a different in resonance lengths of all three harmonics, the first one giving 0.115 m, the second 0.365m and the third 0.61m. The first harmonic would be a single antinode and node in a half-wavelength pattern. One node and two antinodes make up the full-wavelength pattern of the second harmonic. The third harmonic would have a 3/2-wavelength pattern consisting of two nodes and three antinodes. Page 7 of 8

Conclusions: To sum up, this experiment examined the complexities of transverse and longitudinal standing waves. Our goal was to present a thorough investigation of wave phenomena by measuring the linear mass density of a stretched string which was (6.65x10 -4 ± 7.04x10 -5 ) kg/m, computing the standing transverse wave speed in a rope calculated to be (27.16 ± 1.451) m/s, and determining the standing longitudinal wave speed in a spring which was 2.75 m/s. Furthermore, the quite accurate measurement of sound speed as (344 ± 5.537) m/s and the study of standing waves in a tube add to a more sophisticated comprehension of these basic ideas and deepen our grasp of the various standing wave mechanisms. Overall, I learned a lot. Page 8 of 8

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