phy 133 lab 9

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Anubrota Majumdar 4/30/2023 PHY 133 Standing Waves Lab Report

INTRODUCTION: - The purpose of this laboratory experiment is to investigate the relationship between the harmonic value (n) and the tension in the string used to create standing waves using a motor and string. Standing waves are created by the superposition of two waves traveling in opposite directions on a string, resulting in nodes and antinodes where destructive and constructive interference occur, respectively. The speed of the wave on a string can be described by the equation v = √(T/ μ), where T is the tension in the string and μ is the linear mass density, which is given by the mass of the string divided by its length. The relationship between velocity, wavelength, and frequency of a wave can be expressed as v = λ f, where λ is the wavelength and f is the frequency. Changing any of these variables will result in a change in the speed of the wave. If there is a change in linear mass density, the speed of the wave will also change. When a wave encounters a barrier, the wave cannot be transmitted, and all of the energy in the wave is reflected, becoming a reflected wave. If the wave runs into a barrier, the wave will not continue, and the energy will be reflected. If the conditions are right, such as when the incident wave interferes with the reflected wave, a standing wave can result. The nodes of the standing wave occur at points of maximum destructive interference, while the antinodes occur at points of maximum constructive interference. A standing wave appears as if the wave is not moving in either direction. During the experiment, a motor was used to create waves on a string, and the tension in the string was varied to create standing waves at different harmonic values. The tension in the string was measured using a force sensor, and the harmonic value was calculated based on the length of the

string and the wavelength of the standing wave. The period of the wave was also measured using a timer. The collected data was used to investigate the relationship between harmonic value and tension in the string. The results of the experiment showed that the tension in the string increased as the harmonic value increased, indicating a direct relationship between the two variables. In conclusion, the laboratory experiment demonstrated the relationship between the harmonic value and tension in a string used to create standing waves. FORMULAS USED: -  Wave velocity formula: v = ƒλ where v is the velocity of the wave, ƒ is the frequency of the wave, and λ is the wavelength of the wave.  Wave speed on a string formula: v = √(T/μ) where v is the speed of the wave, T is the tension in the string, and μ is the linear mass density of the string (mass per unit length).  Linear mass density formula: μ = m/L where μ is the linear mass density, m is the mass of the string, and L is the length of the string.

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 Harmonic frequency formula: ƒn = nv/2L where ƒn is the frequency of the nth harmonic, n is the harmonic number, v is the velocity of the wave, and L is the length of the string.  Tension formula: T = mg where T is the tension in the string, m is the mass of the hanging weight, and g is the acceleration due to gravity.  Wavelength formula: λn = 2L/n where λn is the wavelength of the nth harmonic, L is the length of the string, and n is the harmonic number. These formulas were used to calculate various properties of the standing waves on the string, such as the frequency, wavelength, velocity, and tension. By analyzing these properties, we were able to observe the relationship between harmonic (n value) and tension in the string.

MATERIALS USED: -  Motor  String  Snell's Law Dish  Table  IOLab device  Force sensor (IOLab)  Eyebolt screw  Glue  Tape  Breadboard  Battery pack  Batteries PROCEDURE:-  Create a barrier around the motor wires and fill it with glue to secure them.  Screw the eyebolt screw onto the force sensor of the IOLab device.  Tie one end of the string around the screw and the other end around the table.  Place the IOLab device on the table and tie a knot around the motor to keep it stable.  Use a Snell's Law Dish near the motor to keep it elevated and prevent it from hitting the table.  Measure the distance between the eyebolt and the motor.  Connect the motor and battery pack to the breadboard, connecting the red wires to each other and the black wires to each other. Insert batteries into the battery pack.  Press record and turn on the battery pack.

 Let the string go slack so the force probe detects 0 at first.  Slowly pull the device away from the motor until a standing wave pattern appears.  Keep the device in that position, turn off the battery pack, and stop recording.  Use the analysis tool to find the mean and sigma tension of that harmonic.  Use the FFT function with a 1024 option to find the peak frequency of the highlighted section.  Create a chart with the following elements: harmonic number (n), force (in newtons), and frequency (in hertz).  Plot the tension force versus the frequency.  Find the slope of the plot to determine the linear mass density of the string.  Compare the experimental value to the known value. OBSERVATIONS: - Reading for harmonic motion where n = 2

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Readings for harmonic motion = 3 Readings for harmonic motion = 4 0 0.01 0.02 0.03 0.04 0.05 0.06 0 20 40 60 80 100 120 140 Tension Force vs (f/n)^2 Harmonic Force(N) Frequency f/n (f/n)^2

Motion 2 0.0485 22.865 11.4325 130.702 3 0.0415 13.856 4.6186 21.332 4 0.003 4.320 1.08 1.1664 So Linear Mass Density = T = 4 L ^2 μ (f/n) ^2 = 0.0022 kg/m. Discussion and Conclusion The results of the lab supported the hypothesis that the tension in a string is inversely proportional to the harmonic or nodes, as shown by the decrease in tension as the harmonic increased. This pattern is demonstrated in the table, where the tension for the second harmonic was 0.0485 N, while the tension for the fourth harmonic was 0.003 N. The frequency also followed this trend, with a decrease in frequency as the harmonica increased. The linear mass density was calculated to be 0.0022 kg/m. However, sources of error could have occurred due to the setup and placement of the motor, which caused the motor to shake and alter the waves. In conclusion, the lab demonstrated the importance of standing waves in illustrating concepts such as harmonics and linear mass density. The results revealed important trends, such as the relationship between tension and the harmonica, which decreased as the harmonic increased. Further experiments could involve finding the mass of the device after calculating the linear mass density and comparing it to the actual mass.

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