Lab report 2

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Northeastern University *

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Physics

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Feb 20, 2024

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Report for Experiment #14 Standing Waves Abstract In this experiment, our main purpose is to study standing waves of a string and in an air column in order to test the wave velocity. These two are the completely related investigations. In investigation 1, we tested the standing waves of a string. We attached a plastic pail at one end of the string to created waves by adding weight (0.2968kg, 0.1678kg, 0.1034kg, 0.0738kg, 0.0603kg, 0.0501kg) and a 120Hz vibrator at another end of the string to show the number of Nodes(3,4,5,6,7,8). We then used weights to measure the tension(2.91N,1.64N,1.01N,0.72N,0.59N,0.49N) and the nodes number and length of the string(1.55m) to measure the velocity(124m/s, 93m/s,74.4m/s,62m/s, 53.14m/s,46.5m/s) and plotted a graph of Tension vs. Velocity^2.In investigation 2, we tested the standing waves in an air column which used 3 different tuning forks with 256Hz, 480Hz, and 512Hz to test the first two resonance of each fork. After this, we length of the first and third resonance to measure the wavelength which are 0.6m, 0.664m, 1.332m.and plotted a graph of wavelength vs. 1/f. Introduction There are many different types of waves, such as sound waves, light waves, and other forms. A wave is made up of successive peaks and troughs, and travels in a certain direction. The motion of the peaks and troughs is divided into two types "transverse" and "longitudinal". "Transverse" means that the motion of the crests and troughs is perpendicular to the direction of wave propagation. "Longitudinal" means that the direction is horizontal. According to the relationship between wavelength, velocity, period and frequency, we can get the following expressions: λ V = = fλ T

The vibrations of the strings produce sound and this sound changes by the different tensions of the strings. The speed of the wave along the string depends on the string tension Fs and the mass per unit length of the string μ. The relationship is 푠 푠푡푟 = The speed of sound in air depends on the atmospheric pressure p, the air density ρ and a constant γ. Since we could not calculate it in this experiment, we put the speed of sound at about 343 m/s. After some background on waves, we will move on to the purpose of our experiment - standing waves. Suppose a train of waves traveling through a medium reaches a boundary beyond which it cannot propagate. At such a boundary, the wave will be reflected back in the direction from which it came. The reflected wave will be superimposed on the incident wave. The result is the formation of a standing wave. When two sine waves of equal amplitude and wavelength move towards each other, their superposition is a stationary sine wave with nodes and antinodes as shown in the F0.1 below. This wave is called a "standing wave". F0.1Displacement of a standing wave for five instances in time F0.2 The three lowest string modes After we understand what a standing wave is, we also need to know how to calculate the wavelength of the standing wave. The key to the formation of a standing wave is that there must be nodes at the two fixed ends of the string. However, as shown in Figure F0.2, they can also have additional nodes at the midpoint. The graph at the top of F0.2 shows the longest wavelength. If there are no additional nodes between the two endpoints of the graph, and the length of the string is L, then the longest wavelength of this standing wave is λ1 = 2L. Thus, we obtain the equation of the wavelength on this string is λ n = 2 n L . Once the wavelength is known, the velocity can be calculated and compared to the predicted value by using Vstring formula. Investigation 1 In investigation 1, the setup that we used are 120 Hz vibrator, 2 rods, 3 clamps, 1 pulley, short rod with string clamp, slotted weights, plastic and metal washers, string, paper clip, plastic pail, meter stick , and digital scale.

F1.1 Setup In this study, we first assembled the instrument as in F1.1. By stretching a rope between two brackets, one end was clamped to the bracket; the other end was tied to a bucket to prevent it from falling. The end with the bucket was hung on a pulley to ensure that the rope could move freely. The tension of the rope is changed by adding or subtracting washers from the bucket to achieve different wavelengths of effect. Then a 120Hz electrical vibrator is used on the left side to barely touch the rope. Each time it touches the string, a small amount of energy is transferred to it. The standing wave is created by adjusting the tension. Before we start the experiment, we turn on the shaker and then change the tension by adding or subtracting weight to find the resonance of the three nodes. At the beginning, the estimated value given in the lab manual was about 900 grams. But when we actually tested it, the 900 grams was much more than the weight that could make the string tremble. So much so that we adjusted it to about 500 grams to clearly see the vibration of the string. Afterwards we made a table to record the number of nodes and the distance between adjacent nodes, and to determine the wavelength to find the error in the wavelength measurement. Tension (Fs=mg) measurements were also recorded. Because of the large difference between the weight needed for our test and the one given by the experiment, we applied the error of gravity instead when calculating the error of tension. According to the following equation from Appendix A, we can calculate the error of gravitational force . The g in the equation of gravitational force is a constant. Thus, the error of the gravitational force should be F δm . The error of velocity and wavelength should use the same equation as the tension. Since the m wavelength is equal to length*2/n, n is a constant which did not have error, error of wavelength should be λ δ L . According to the equation V = fλ , in this investigation, the frequency f is a constant 120Hz which

L did not have a error. The error of the velocity should be V . The equation of the error of V^2 is given in the manual. After finished the calculation, we did the same steps in 4,5,6,7, and 8 nodes. The data we measured is in the table below. T1.1 Properties of wave in string with error for different nodes Nodes 3 4 5 6 7 8 Weight(kg) 0.2968 0.1678 0.1034 0.0738 0.0603 0.0501 Length(m) 1.55 1.55 1.55 1.55 1.55 1.55 error of Length(m) 0.005 0.005 0.005 0.005 0.005 0.005 Distance between two nodes(m) 0.516 0.3875 0.31 0.35 0.2214 0.19375 Wavelength(m) 1.0333 0.775 0.62 0.5167 0.4429 0.3875 error of wavelength(m) 0.003333 0.0025 0.002 0.001667 0.001429 0.00125 Force Tension(N) 2.9086 1.6444 1.0133 0.7232 0.5909 0.4910 error of Force Tension(N) 0.0049 0.0049 0.0049 0.0049 0.0049 0.0049 velocity of string (m/s) 124 93 74.4 62 53.143 46.5 error of velocity(m/s) 0.3871 0.3871 0.3871 0.3871 0.3871 0.3871 velocity squared (m/s^2) 15376 8649 5535.36 3844 2824.163 2162.25 error of velocity squared (m/s^2) 96 72 57.6 48 41.143 36 Then, we plotted the tension F in the string vs. V^2 of the string. F1.2 Tension vs. Velocity^2

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