lab 14

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Report for Experiment #14 Standing Waves Anh Nguyen Lab Partner: Ishaan Desai TA: Hung-Yi Ly 11/29/2022 Abstract

Introduction In this experiment, we will study the properties of waves. We will observe waves in the air and on a string, we also study the relationship between string tension and wave velocity. With the data collected, we will calculate sound velocity. There are 2 types of waves, transverse waves move up and down just like waves in the oceans, longitudinal waves move back and forth between regions with different pressure like sound in the air. A snapshot of transverse wave is shown below. Amplitude A is the distance between the wave peak and 0. The distance between 2 peaks is called a wavelength. The time it takes to move from 1 peak to another will be called a period. And we know that velocity is distance over time, substitutes in the wave properties, we will have the equation (1) Where (2) Vibration can cause sound, just like a guitar string. It causes waves in the surrounding air, when a guitar is tuned, it is adjusted to play the right note. This is because the frequency depends on the tension and the material of the string, and so is velocity. Velocity can be found using tension force Fs and mass per unit length μ of the string. (3)

Sound in the air is a little different, it can be calculated using the formula (4) Standing waves are waves that as it reach the end and reflects back, the original one and the reflected one are superposition, in which retains its identity and travels independently of any other wave. The sum of these 2 waves will result in a standing wave. The parts on waves that are 0 are called nodes, and the maxima points are antinodes. There are modes standing waves can form, but they all must have node at the 2 fixed end like the image below Wavelength can be calculated with formula below (5) Where L is the length of the string and n is the number of antinodes. Error of propagation to find error is (6)

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Investigation 1 In this investigation, we will observe standing waves on a string. We will use a fishing line as the medium We will set up the experiment like above, the string will be held down by a weight bucket on an apparatus. The vibrator will produce waves along the string. We will put weight on the bucket to find standing wave and record the data. We will create standing waves with 3,4,5,6,7,8 nodes. For each standing wave, we will record the weights, the number of nodes, and the distance between adjacent nodes. We average the distance between adjacent nodes and this will be our wavelength. Because wavelength is the distance in which the wave repeats itself. With the weight data, we can find the tension force and its error, the error will be how much weight will it take to make the standing wave disappear. node number avg distance (m) error distance (m) wavelength (m) error wavelength (m) force tension (N) error force tension (N) 3 0.735 0.0005 1.47 0.001 8.829 0.6867 4 0.502 0.0005 1.00 0.001 4.414 0.1962 5 0.405 0.0005 0.810 0.001 2.207 0.1658 6 0.323 0.0005 0.646 0.001 1.104 0.1550 7 0.267 0.0005 0.534 0.001 0.5518 0.1540 8 0.220 0.0005 0.440 0.001 0.2759 0.1265 Next, we will calculate the velocity using equation (1), we have our wavelength data, and the vibrator frequency is 120 Hz, its error can be found with equation (6). We also find v square and its error, the error can be calculated using the relation below.

node number velocity (m/s) error velocity (m/s) velocity squared ((m/s)^2) error vsquared ((m/s)^2) 3 176.4 0.12 31116.96 42.336 4 120.4 0.12 14496.16 28.896 5 97.2 0.12 9447.84 23.328 6 77.52 0.12 6009.3504 18.6048 7 64.08 0.12 4106.2464 15.3792 8 52.8 0.12 2787.84 12.672 Then we plot tension force vs velocity square The slope of this graph will be the mass per unit length of the string, which is 3218.3 kg/m. And the given data is .32 g/m. Our values are almost identical. Investigation 2 In this investigation, we will observe standing waves in the air. Vibration in air can cause sound like a flute. We will use a hollow cylinder partially filled with water to recreate this. Let ’ s take a look at the image below y = 3218.3x + 2003.9 R² = 0.9931 0 5000 10000 15000 20000 25000 30000 35000 0 1 2 3 4 5 6 7 8 9 10 Velocity Squared ((m/s)^2) Tension (N) Velocity Squared Vs Tension Force

A tuning fork will be held right above the tube to produce sound, as the sound reaches the water level it will bounce back. We will try to find the length of the air column that will produce standing waves. Before doing this we estimate the length of the air column using the formula (7) below (7) To find the wavelength, we will be using equation (1), with speed of sound and the frequency of each given tuning fork. With the tuning fork held at the top of the tube, we adjust the water level above the estimated value and start lowering it. When standing waves occur, there will be a sudden increase in sound intensity, and we record the length at which it happens. Then we raise it again to check for that location and lower it one more time, and we take the average of all 3. We will have 3 measurements for each n. We repeat the process to find the data for n=1, 3, 5. And repeat the whole thing with 2 more different tuning forks. Some tuning fork we can only find n=1 and 3 because the tube isn ’ t long enough. For each fork, we find the wavelength using equation (8) below (8) - 256 hz 480 hz 512 hz 1st resonance avg height (m) 32.5 16.7 16.0 Error 1 (m) 0.0500 0.140 0.170 2nd resonance avg height (m) 100. 52.7 50.9 Error 2 (m) 0.130 0.680 0.810 3rd resonance avg height (m) 89.0 83.2 Error 3 (m) 1.21 1.18 Wavelength (cm) 135.5 71.9 69.8 Lamda Error (cm) 0.261 1.43 1.68 Then we plot lamda with 1/f y = 36961x - 7.3909 R² = 0.9491 0 20 40 60 80 100 120 140 160 180 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 Wavelength (cm) Period (s) Wavelength vs Period

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Lamda over 1/f is equal to lamda multiplied by f, so the slope of this graph will equal to speed of sound according to equation (1). Our calculated speed of sound is 36961 cm/s and the actual value is 343 m/s. Our data isn ’ t in the error range because for the tuning fork with low frequency, we have to keep hitting it which causes it to not be held constant at the top of the tube. This causes the reading to be off. Conclusion In this experiment, we learn about the properties of waves. In investigation 1, we observe standing waves on string and study the relationship between tension and velocity. With the obtained data, we used it to find the mass per unit length of the string which is 3218.3 kg/m, the given data was .32 g/m. The result for investigation 1 was accurate. In investigation 2, we observed standing waves in air, we using sound and a hollow tube to created standing waves. We find the length of air that will create standing waves, and from the data we calculate the speed of sound which is 36961 cm/s. Compared to the actual value 343 m/s, our value isn ’ t within the range of error. A way to make this experiment better is to replace the tuning fork with a device that created frequency so that we can hold it constant above the tube without having to hit it. Questions 1 322.4 m/s 2 520 N 3 344.8 Hz 4 .997 m 5 the resonance frequency increase as the humidity increase Acknowledgments References

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