PHYS 130 Lab 4 (4)

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Feb 20, 2024

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Introduction: Most of us communicate verbally, made possible by sound waves vibrating the surrounding air. Studying the properties of sound waves have allowed us to enjoy auditory entertainment, like symphonies and concerts. However the convenience of sound waves is not limited to the earth; black holes tend to emit sound waves across the cosmos which we can measure to determine the pitch of these interstellar enigmas, and discover more about the universe we live in [3, 4]. The objective of this lab is to measure the speed of sound in air using two different methods, the first being standing waves in a cylindrical pipe which will help us learn about resonance and also

understand the nodes and antinodes of a wave [1]. With one end of the pipe open while the other is closed, standing waves can be created through the emission of certain frequencies near the open end of the tube [2]. The sound will form standing waves when the tube is resonant with the frequency [1]. In the case of our experiment, the frequencies used to generate these standing waves were produced using four different tuning forks each corresponding to a specific musical note: C, B, A, or G where the frequencies are 512 Hz, 480 Hz, 426.6 Hz, and 384 Hz respectively. The standing wave consists of a series of nodes and antinodes, where the nodes are positions where the air molecules are stationary and antinodes are positions when the air molecules vibrate at a maximum amplitude [1]. At the opening of the pipe exists an antinode that is located a small distance, defined as the end correction in this lab, outside the pipe [1]. By determining the resonance points for the frequencies of the four different musical notes we can use the linearized equation Eq 4.1 ? ? = ? ? (2?−1) 4? − ? 0 , ? = 1, 2, 3... to solve for the speed of sound, in meters per second, where is the distance in meters inside the ? ? ? ? chamber where an antinode is heard, is the ordinal number of the resonance, is the frequency of the ? ? tuning fork, and is the end correction of the antinode at the opening of the tube. ? 0 Our second method in calculating the speed of sound utilizes the Phyphox app. Using the acoustic stopwatch feature we measure the time it takes between two acoustic events, in our case two claps, in seconds over a set distance between two phones. The equation Eq 4.2 ? ? = 2?/∆? can be used to determine the speed of sound, where is the distance between the two phones in meters ? and is the difference of time in seconds of each of the phone’s readings. ∆? The theoretical value for the speed of sound can be calculated using the following: Eq 4.3 ? ? = ? ?, 0 𝑇 273.15 𝐾 where is the speed of sound in dry air at 0º C (273.15 K) measured at 331.1 m/s, and is the ? ?, 0 𝑇 temperature of the room in Kelvin [1]. We can also determine the theoretical approximation of the end correction using ? 0 Eq 4.4 ? 0 ≈ 0. 6𝑅 where is the radius of the pipe. 𝑅 Methods: We set up the apparatus for our first method as shown in Fig A.1 in the appendix. After striking the rubber rod with one of the tuning forks, we held the vibrating fork up to the open end of the cylindrical pipe and slowly moved the end of the plunger back until we heard an amplified tone. Using a meter stick we measured the length of the end of the plunger at the point where the loud amplification was heard; this corresponds to the first resonance point, . Striking the same tuning fork, again we held it up to the open ? 1 end of the tube and pulled the plunger backwards until another amplified tone was heard and measured the length of the end of the plunger, which corresponds to the second resonance point . We continued ? 2 the previous steps for the third resonance point and repeated the entire process for the other three ? 3 tuning forks, measuring two or three resonance points for each fork. After completing this procedure for each fork, we repeated the experiment a second time, switching roles for who striked the rod and moved the plunger. After the second trial we used a small ruler to measure the length of the tube’s diameter in centimeters. We set up our experiment for the second method as shown in Fig A.2 in the appendix. To measure the speed of sound, we used the acoustic stopwatch feature in the PhyPhox app, adjusting the

threshold so the stopwatch would not be triggered by noise softer than a clap. Setting the two phones a fixed distance apart between two walls, my lab partner clapped to start the timers, the stopwatch on my end starting slightly later. A few seconds later I clapped to stop the timers and we recorded the times for each phone, the difference being the time it took for the sound to reach the second phone. We recorded four trials of this experiment. We used an infrared thermometer to measure the temperature used to calculate the theoretical speed of sound in Eq 4.3. Results: Using Eq 4.3 we calculated the theoretical value for the speed of sound when the temperature of the room, , was measured to be (20.8 ± 0.1)ºC which in Kelvin would be (294.0 ± 0.1) K. Calculating 𝑇 through Eq 4.3 and using the general error formula for uncertainty δ? ? = ( ∂? ? ∂𝑇 ) 2 δ𝑇 2 = ( ? ?,0 2·273.15 𝑇 273.15 ) 2 δ𝑇 2 we get = (343.37 ± 0.06) m/s (details of this derivation can be found in Section A.1 in the appendix). ? ? The diameter of the pipe was measured to be (0.038 ± 0.001) m, the uncertainty being the smallest increment of measure on our ruler. After solving the pipe’s radius to be (0.019 ± 0.001) m, we can use the Eq 4.4 and the uncertainty formula δ? 0 = ( ∂? 0 ∂𝑅 ) 2 δ𝑅 2 to determine the theoretical end correction to be (0.0114 ± 0.0006) m (see appendix Section A.2 for ? 0 the full derivation of ). ? 0 Our first method in calculating the experimental value of uses Eq 4.1: ? ? Distance (Dn) ( ± 0.001 m) 512 Hz 480 Hz 426.6 Hz 384 Hz n = 1 0.153 0.167 0.199 0.207 n =2 0.491 0.523 0.598 0.659 n =3 0.831 0.893 0.984 N/A Table 4.1.1: Data table of the distance in the chamber from the open end of the pipe to the top of the plunger in meters as well as the frequency of the tuning fork in hertz for the first trial. The values n = 1, 2, and 3 correspond to the first, second, and third resonance points respectively. The uncertainty for Dn is the smallest measurement on the meter stick. Distance (Dn) ( ± 0.001 m) 512 Hz 480 Hz 426.6 Hz 384 Hz n = 1 0.158 0.167 0.201 0.209 n =2 0.494 0.521 0.585 0.648 n =3 0.828 0.887 0.991 N/A

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Table 4.1.2: Data table of the distance in the chamber from the open end of the pipe to the top of the plunger in meters as well as the frequency of the tuning fork in hertz for the second trial. The values n = 1, 2, and 3 correspond to the first, second, and third resonance points respectively. The uncertainty for Dn is the smallest measurement on the meter stick. The table of values for and a sample calculation can be found in Table A.3 in Section A.3 of the (2?−1) 4? appendix. Fig 4.1: This graph shows the distance of each resonance point from the open end of the pipe to the top of the plunger (in meters) plotted against the linearized x-values (in seconds). The data for the 2?−1 4? x-values are from the last column of Table A.3 found in the appendix and the data for the y-values are from Table 4.1.1 and 4.1.2. The data is fit to the equation where and ? = ?? + ? ? = (343 ± 1) ?/? ; the slope and intercept of the graph were found using the LINEST function in ? = (− 0. 010 ± 0. 003) ? Google Sheets. The slope of our graph is equivalent to from Eq 4.1, therefore our experimental value for the ? ? speed of sound is . The y-intercept is equivalent to , the calculated experiment value (343 ± 1) ?/? − ? 0 being , therefore is . (− 0. 010 ± 0. 003) ? ? 0 (0. 010 ± 0. 003) ? Our second method involves Eq 4.2: The table containing our data of all four trials and a sample calculation can be found in Section A.4 in the appendix. Out of the four trials, we picked the value of that was closest to the theoretical speed of ? ? sound for our comparison which concluded to be (416 ± 5) m/s, the error equation used to solve for the uncertainty being δ? ? = ( ∂? ? ∂? ) 2 δ? 2 = . ( 2 ∆? ) 2 δ? 2

The detailed derivation of this value can also be found in Section A.4. Discussion: The measured experimental values for the speed of sound for method one and two is (343 ± 1) m/s and (416 ± 5) m/s respectively. Comparing each to the theoretical speed of (343.37 ± 0.06) m/s using the formula we get (0 ± 1) m/s for method one and (73 ± 5) m/s for (? ??? − ? ?ℎ?? ) | | ± (δ? ??? ) 2 + (δ? ?ℎ?? ) 2 method two. The experimental speed of sound calculated for the pipe portion of our lab is within one error interval with the theoretical speed, therefore these values are in good agreement with each other. However, our experimental value calculated for the speed of sound using the Phyphox app in method two is in poor agreement with the theoretical value as they are more than three error intervals apart. The calculated value for the end correction in our lab was (0.010 ± 0.003) m, the theoretical ? 0 end correction being (0.0114 ± 0.0006) m. Using the comparison formula we get a value of (0.001 ± 0.003) m. We can determine that our experimental and theoretical values are in good agreement as they are within one error interval. Although our values are in good standing for our first method of calculating the speed of sound, minor modifications could be made to improve the data collection process of the experiment. For example, a decibel meter could be used to ensure that the resonance we heard was indeed the loudest sound heard at that point, instead of relying on our interpretation of the sound. The inaccuracy of method two could be explained by the equipment used for the experiment. Because method two solely relied on the efficiency of the Phyphox app’s acoustic stopwatch, systematic errors can be considered when explaining the imprecision in our experimental speed of sound. The timer could have been triggered by a sound other than our claps, as we performed this experiment in a hallway between lecture rooms, which would affect the difference of time between both phones. An improved version of this experiment could be done in a silent setting, where the noise that triggers the stopwatch is limited to the one intended by the experimentee. Another source of error could be human error, where the measurement of the distance between the two phones was not accurate as we used one meter stick to measure a length greater than a meter. To correct this, a more precise method to measure the distance (such as a tape measure) could improve the results of the experiment. Conclusion: We determined the speed of sound in air using two methods. The first utilized a cylindrical pipe with one end closed, striking tuning forks of various frequencies to create standing waves by aligning the vibrating forks to the open end of the tube, and measured the resonance points of each fork’s note. The measured value for the speed of sound for this portion was (343 ± 1) m/s which was in good agreement with the theoretical speed of (343.37 ± 0.06) m/s as they were within one error interval. For the second method of the experiment, we used the acoustic stopwatch in the Phyphox app, measuring the speed of sound as (416 ± 5) m/s which was in poor agreement with the theoretical speed as they were more than three error intervals apart. The values of our end corrections were in good agreement as (0.010 ± 0.03) m and (0.0114 ± 0.006) m are within one error interval apart. References: [1] Lab Manual PHYS 130. Edmonton: University of Alberta, Department of Physics. I used this reference for the points labeled throughout and the equations used in the lab as well as for the overall structure and articulation of the report. [2] Open & Closed Pipe (Physics): Differences, Resonance & Equation . (2021, April 29). Sciencing. https://sciencing.com/open-closed-pipe-physics-differences-resonance-equation-13721570.html

[3] Interactio. (2022, June 14). The importance of sound quality. Interactio . https://www.interactio.io/post/importance-of-sound-quality#:~:text=Sound%20is%20a%20crucial%20aspe ct,a%20lot%20about%20the%20message . [4] Can you hear a black hole? (2013, October 29). NASA Blueshift. https://asd.gsfc.nasa.gov/blueshift/index.php/2013/10/29/maggies-blog-can-you-hear-a-black-hole/ Speed of Sound Sample Lab Report on eClass. I used this resource to guide me on how to articulate the captions for each of my tables and graphs, as well as overall flow of the report. Appendix: Fig A.1: Labeled image of our equipment and setup for the first method of our experiment.

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Fig A.2: Labeled image of the recreation of our equipment and setup of method two of our experiment using the Phyphox app. The setup for our actual experiment was too difficult to capture as the distance was too long to document efficiently so the image above is an adequate reproduction of our real test. Section A.1- Derivation of the Theoretical Value of C s : To convert our temperature from (20.8 ± 0.1)ºC to Kelvin: 𝑇 𝐾 = 𝑇 º? + 273. 15 𝑇 𝐾 = 20. 8 + 273. 15 = 294. 0 𝐾 We kept the uncertainty for the temperature in Kelvin the same as the temperature in Celsius (the smallest increment of measurement on the infrared thermometer), giving us a final value of (294.0 ± 0.1) K. Using this temperature we can calculate for the theoretical speed of sound ? ? = ? ?, 0 𝑇 273.15 𝐾 = 331. 1 ?/? 294.0 𝐾 273.15 𝐾 = 343. 3714... ?/? and it’s uncertainty δ? ? = ( ∂? ? ∂𝑇 ) 2 δ𝑇 2 = ( ? ?,0 2·273.15 𝑇 273.15 ) 2 δ𝑇 2 = ( 331 2·273.15 294.0 273.15 ) 2 (0. 1) 2 = 0. 0584... ?/? to get a theoretical value of (343.37 ± 0.06) m/s. Section A.2- Derivation of the Theoretical Value of x 0 : ; where is the diameter of the pipe, (0.038 ± 0.001) m, and is the radius. ? 2 = 𝑅 ? 𝑅 0.038 2 = 0. 019 ? We set the uncertainty for the radius the same as the uncertainty of the diameter, resulting in a radius of (0.019 ± 0.001) m. Using the radius, can be calculated by ? 0

? 0 ≈ 0. 6𝑅 ≈ 0. 6(0. 019) ≈ 0. 0114 ? and it’s uncertainty δ? 0 = ( ∂? 0 ∂𝑅 ) 2 δ𝑅 2 = (0. 6) 2 δ𝑅 2 = (0. 6) 2 (0. 001) 2 = 0. 0006 ? to get a theoretical value of (0.0114 ± 0.0006) m. Section A.3- Linearized x-values of Frequency Frequency (Hz) n x-values (s) 512 1 0.000488 512 2 0.00146 512 3 0.00244 480 1 0.000521 480 2 0.00156 480 3 0.00260 426.6 1 0.0005860 426.6 2 0.001758 426.6 3 0.002930 384 1 0.0006510 384 2 0.001953 Table A.3: Data table of the resonance point, n, corresponding to each tuning fork frequency in hertz. The x-values column consists of the linearized calculated values plotted on the x-axis of Fig 4.1 using the equation . (2?−1) 4? Sample calculation of x-values column in Table A.3: (2?−1) 4? = (2×1−1) 4(512 𝐻?) = 1 2048 𝐻? = 0. 000488281 ? = 0. 000488 ?

Section A.4- Method two (Phyphox App Portion) Data and Calculations Phone A (s) Phone B (s) ∆t (s) C s (m/s) C s (m/s) δ 2.018 1.997 0.021 436 4 2.558 2.538 0.020 458 5 2.936 2.918 0.018 509 6 3.596 3.574 0.022 416 5 Table A.4: Data table of the recorded time in seconds using the acoustic stopwatch feature in Phyphox of each phone, the difference between the time readings ∆t, and the calculated speed of sound c s as well as its uncertainty. The distance variable (the distance between the two devices) in the Eq 4.2 used to determine c s was a fixed length of (4.58 ± 0.05) m, where the uncertainty was set to 0.05 meters due to the us having to approximate the distance of each meter between the phones with the use of only one meter stick. Sample Calculations for Table A.4: ∆? = 𝑃ℎ??? ? − 𝑃ℎ??? ? = 2. 018 − 1. 997 = 0. 021 ? ? ? = 2?/∆? = 2(4. 58)/0. 021 = 436. 19 ?/? The uncertainty for c s can be found using the general error formula: δ? ? = ( ∂? ? ∂? ) 2 δ? 2 = ( 2 ∆? ) 2 δ? 2 = ( 2 0.021 ) 2 (0. 05) 2 = 4. 761... ?/? The c s value used in the comparison will be (416 ± 5) m/s.

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PHYS 130 Lab 4 (4)

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