DianaSooknauthLabReportExp14

Report for Experiment #14 Standing Waves Diana Sooknauth Lab Partner: Nola Hallemeier TA: Cort Thoreson November 28 th , 2023 Abstract The purpose of this experiment was to observe standing waves both on a string and in the air. We were looking to see the relationship between wave velocity, tension, frequency, and wavelength. In order to do so, we first used an apparatus that allowed us to see waves moving on a string that was attached to a bucket, creating different tensions strengths as we added different amounts of weight. From this investigation, we were looking to find the mass per unit length of the string which was initially given to us as .32 g/m. The value we found was 0.0003 ± 2.15 E-05 kg/m which is what we expected. In the second investigation, we measured the resonances of sound waves in an air column with a speaker playing different frequencies at the end. The objective of this investigation was to find the speed of sound. The given value for the speed of sound is 343 m/s and the value we found was 317.4 ± 13.81 m/s. This was not within the uncertainty range for the value we expected.

Introduction Waves are nature ’ s way of transporting energy. Waves in nature can exist as sound waves, light waves, and even matter waves. A wave consists of crests and troughs that travel in a certain direction. There are two main types of waves: transverse and longitudinal. Transverse waves are waves which move in a direction perpendicular to the medium of the wave. For example, on the ocean surface, crests and troughs are formed by the up and down motion of the water. These waves are perpendicular to the ocean surface, making them transverse. Longitudinal waves occur when the particles of a medium move back and forth parallel to the direction of the wave. An example of longitudinal waves are sound waves in air. Near the crests of the wave, individual air molecules will move towards each other, building up the pressure in that region. Near the troughs the individual air molecules will move away from each other. Longitudinal waves do not create any overall motion of the medium, and instead there are vibrations back and forth of its individual elements. A transverse wave of amplitude A and wavelength λ traveling along a string in the positive x- direction can be seen in Figure 1 below. Figure 1: A transverse wave of amplitude A and wavelength travels along a string in the positive x. direction. Source: Macmillan Learning, Introductory Physics Manual by Batishchev The speed of the wave of the motion is given by 𝑣 = ∆𝑥/∆𝑡 . The time interval during which the peak at 𝑥 1 moves all the way to 𝑥 3 (the wavelength) is called the period T . The period is the time for one complete oscillation of a point on the string. The inverse of this, the number of oscillations of a point in one second, is called the frequency, f . From this information, we see two important relations: 𝑣 = ? 𝑇 = ?? (1) ? = 1 𝑇 (2)

Figure 2: The apparatus for Investigation 1. Source: Macmillan Learning, Introductory Physics Manual by Batishchev To set up the apparatus, we first stretched a string between two stands. The string had a mass per unit length of 0.32 g/m, which was given to us. One end of the string was clamped to the stand, and the other was connected to the plastic bucket on the pulley. We then turned on the vibrator and searched for the three-node resonance. We began with having 900 g in the bucket. We started adding slight weight to the bucket by adding one steel washer at a time until resonance occurred. We know we have reached resonance when we touched a node and there was no oscillation of the string. We then made a table where we recorded the number of nodes and the distances between adjacent nodes. We then calculated the average distance between adjacent nodes and calculated the wavelength along with its error using the following equations: ? = 2 × 𝑥 ??𝑔 (5) 𝛿? = √ 𝛿𝑥 ??𝑔 (6) We then recorded the measurement of the tension in the string, as well as its error. The tension error is found by seeing how much weight it takes to make the standing wave disappear. We calculated the tension by using the equation: 𝐹 2 = ? ???𝑘?? × ? (7) We then calculated wave velocity, 𝑣 ???𝑖?𝑔 , using Eq.1 with our measured wavelengths and the vibrator frequency, which was given as 120 Hz. We then found the error by using the equation: 𝛿𝑣 ???𝑖?𝑔 = ?? ? × 𝑣 ???𝑖?𝑔 (8)

We then found (𝑣 ???𝑖?𝑔 ) 2 by squaring 𝑣 ???𝑖?𝑔 and we found its error by using the equation: 𝛿(𝑣 ???𝑖?𝑔 ) 2 = 2𝑣 ???𝑖?𝑔 𝛿𝑣 ???𝑖?𝑔 (9) We then repeated these steps for 4,5, and 6 node cases. We then made a plot of the tension in the string vs. the velocity squared in the string. We are able to obtain a measured value of the mass per unit length of the string, ? by looking at Eq. 3. If we solve for ? , we can see that ? is the slope of the graph. The graph created can be seen in Figure 3. Figure 3: Plot of the Tension in the String vs. Velocity Squared in the String for Investigation 1. The value we obtained for the mass per unit length of the string is 0.0003 ± 2.15 E-05 kg/m. The error was calculated using the IPL Straight Line Fit calculator. We can see this value from the slope on the graph, y= 0.0003x -0.7819. This value agrees with the given value of mass per unit length which is 0.32 g/m. If we convert the grams to kilograms, we see that we have obtained the exact value within error. The data collected can be seen in Table 1 and Table 2 below. # of nodes distance 1 (m) distance 2 distance 3 avg distance (m) Error (m) 3 0.7700 0.7800 0.7750 0.0010 4 0.5200 0.5400 0.5200 0.5267 0.0010 5 0.4100 0.3900 0.4000 0.3900 0.3975 0.0010 6 0.3400 0.3200 0.3300 0.3200 0.3100 0.3240 0.0010

We then found the distance between the first and second resonance, and determined the wavelength by using the equation: ? = (𝐿 3 4 − 𝐿 1 4 ) × 2 (11) We then made an estimate of the error in the wavelength measurement. By using the equation: 𝛿? = √ (𝐿 3 4 ) 2 + (𝐿 1 4 ) 2 (12) We then repeated the procedure of finding the lengths at which the resonance was for the frequencies of 805 Hz and 1061 Hz. The wavelengths we found were 0.6933 ± 0.0122 m for 479 Hz, 0.4333 ± 0.0122 m for 805 Hz, and 0.3267 ± 0.0122 m for 1061 Hz. We then plotted ? vs. 1/? which can be seen in Figure 4. Figure 4: Plot of the Wavelength vs. Period for Investigation 2. We found the slope to be y=317.4 ± 13.81x. We determined the error by using the IPL Straight Line Fit calculator. If we look at Eq. 1 and Eq. 2 we can see that there is a relationship between the speed of sound, wavelength, and the period/frequency. The slope we obtained is supposed to represent the speed of sound. The value of the speed of sound we were given, 343 m/s, is not within the error and is smaller than the value we expected. This could have occurred due to experimental error or random error. We were

working with equipment that isn ’ t highly precise, and this could have contributed to a error in our measurement and our data. The data we collected for Investigation 2 can be seen in the table below. Figure 5: Data collected for Investigation 2. Conclusion The goals of the experiment were to study standing waves of a string, examine the relationship between string tension and wave velocity, study standing waves in an air column, and measure the sound velocity. We were able to meet our goals by collecting data about standing waves on a string. In the first investigation, we used an apparatus to observe the correlation between standing waves and the tension in the string. What we noticed was that the less tension in the string, the more nodes there were. From the data we collected, we were able to find the mass per unit length of the string which was 0.0003 ± 2.15 E- 05 kg/m and what we expected considering the given mass per unit length of the string was .32 g/m. In the second investigation, we observed standing waves in air. We used an air column with a movable plug and a speaker emitting a chosen frequency. We measured where the resonances of the waves were and used that data to determine the speed of sound from a graph. The value we obtained from our slope was 317.4 ± 13.81 m/s which was not within the error for the given value of 343 m/s. This experiment allowed us to observe the physical patterns of waves and also hear waves through sound. Some error that we may have faced caused our results in Investigation 2 to come out not as we expected could have been due to the equipment we used. We were not using precise equipment and the air column specifically was facing some issues where the plug would get stuck in some places, or the whole 479 Hz d1 (m) d2 d3 avg error L1/4 0.1800 0.1600 0.1700 0.1700 0.0087 L3/4 0.5100 0.5100 0.5300 0.5167 0.0087 L5/4 0.8800 0.8900 0.8900 0.8867 0.0087 805 Hz d1 (m) d2 d3 avg error L1/4 0.0700 0.0800 0.0800 0.0767 0.0087 L3/4 0.2900 0.2900 0.3000 0.2933 0.0087 L5/4 0.4900 0.5100 0.5100 0.5033 0.0087 1061 Hz d1 (m) d2 d3 avg error L1/4 0.0500 0.0500 0.0500 0.0500 0.0087 L3/4 0.2100 0.2100 0.2200 0.2133 0.0087 L5/4 0.3800 0.3800 0.3900 0.3833 0.0087 wavelength error 1/f 0.6933 0.0122 0.0021 0.4333 0.0122 0.0012 0.3267 0.0122 0.0009

Appendix A Excel sheet with all of the collected data: Physics2.Lab6.xlsx