Lab14_-_Standing Waves on a String

PHY 202 Name: ____Gabrielle Thompson________________________ Standing Waves Objective In this series of experiments, the resonance conditions for standing waves on a string will be tested experimentally. Theoretical Background Consider an elastic string. One end of the string is tied to a rod. The other end is under tension by a hanging mass and pulley arrangement. The string is held taut by the applied force of the hanging mass. Suppose the string is plucked at or near the taut end. The string begins to vibrate. As the string vibrates, a wave travels along the string toward the fixed end. Upon arriving at the fixed end, the wave is reflected back toward the taut end of the string with the same amount of energy given by the pluck, ideally. At certain frequencies (the amount of plucking per unit of time), the interaction between the waves will produce a resulting wave pattern called a standing wave . Standing waves occur when there is no left or right motion of the wave patterns. Frequencies producing standing waves are resonant frequencies . Figure 1 shows the first and second set of standing wave patterns for a string held taut. Figure 1: Standing Waves on a String As shown on Figure 1, the stationary string positions in the standing wave pattern are known as nodes , while the positions with maximum amplitude in the pattern are known as antinodes . The first standing wave pattern is referred to as the fundamental or first harmonic of the string. In

this pattern, there are no nodes between the two ends of the string (the fixed ends are nodes for the pattern, though they are generally disregarded since they are present in all the patterns). The next standing pattern, with one node between the two ends, is known as the second harmonic . Standing waves with more nodes between the ends are higher harmonics (i.e. third harmonic, forth harmonic, etc.). Notice that the distance between two nodes represents half a wavelength, λ, of the standing wave, as shown as in Figure 1. The wavelength can be related to the length bounded by the two ends. This is the length of the string undergoing propagation, L . For the fundamental, one half of the full wavelength is contained within the bounds. For the second harmonic, one full wavelength is contained within bounds. The third harmonic would have two nodes, which would mean there would be one and a half wavelengths between the string ends. The progression of wavelengths can be expressed by the following mathematical equation: λn = 2 L n , n = 1 , 2, 3, … In this equation, λ n is the wavelength of the standing wave, L is the length of the string bounded by the left and right ends, and n is the standing wave pattern, or harmonic, number. For the fundamental, n would be one, for the second harmonic, n would be two, etc. The resonant frequency can be found by using the relationship between the wavelength and the frequency for waves as shown in the following equation: v = λ f In this equation, v is the (phase) velocity of the waves on the string, λ is the wavelength of the standing wave, and f is the resonant frequency for the standing wave. For waves on a string the velocity of the waves is given by the following equation: v = √ T μ In this equation, v is the velocity of the waves on the string, T is the tension in the string, and μ is the mass density of the string given by the total mass of the string m divided by the total length of the string. These relationships will be tested experimentally in this series of experiments.

Part II – Effect of tension on wavelength 1. Pause the display. Pick a random frequency and record it in Table 2. 2. Use the same linear density as in Part I and record it in Table 2. 3. Slide the tension down to its minimum value, 10 N. 4. Record length of string and recheck measured length with your ruler, recording this as well. 5. Start the simulation and slowly increase the tension to find the first resonance point. Record that tension value in Table 2. 6. Pause the display when the string is near maximum displacement and count the number of string displacements. This should be the harmonic count. If you have just one vibration between the ends, that would be n = 1, the fundamental wave. If you have two up-and-down displacements, that would be n = 2, and so on. Record the count in Table 2. 7. With the simulation paused, with a ruler measure the wavelength of the wave – the distance from one of the crests to the crest ahead of it (or trough and the trough ahead of it). Record the length in Table 2. 8. Continue to increase the tension, finding additional tension values that cause the wave to oscillate at a maximum displacement, recording the tension, count of displacements, and length between waves in Table 2 until you have run out of tension values (100 N). Each subsequent wave pattern may not be as large in displacement as the first but will be the largest at that particular frequency range. You may or may not fill the table, depending on initial conditions. Part III – Effect of linear density on wavelength 1. Pause the display. Keep the frequency from Part II and record it in Table 3. It may be necessary to hit the refresh arrows on the top right. 2. Use the same Tension as in Part I and record it in Table 3. 3. Slide the linear density down to its minimum value. 4. Record length of string and recheck measured length with your ruler, recording this as well.

5. Start the simulation and slowly increase the linear density to find the first resonance point. Record that linear density value in Table 3. 6. Pause the display when the string is near maximum displacement and count the number of string displacements. This should be the harmonic count. If you have just one vibration between the ends, that would be n = 1, the fundamental wave. If you have two up-and-down displacements, that would be n = 2, and so on. Record the count in Table 3. 7. With the simulation paused, with a ruler measure the wavelength of the wave – the distance from one of the crests to the crest ahead of it (or trough and the trough ahead of it). Record the length in Table 3. 8. Continue to increase the linear density, finding additional linear density values that cause the wave to oscillate at a maximum displacement, recording the linear density, count of displacements, and length between waves in Table 3 until you have run out of linear densities. Each subsequent wave pattern may not be as large in displacement as the first but will be the largest at that particular frequency range. You may or may not fill the table, depending on initial conditions.

Table 2 Frequency, f = _____100.00_____________ Hz Mass per unit length, μ = ___ 3.15 × 10 − 3 ________ kg/m Length of string, L = ____4.00_______________ m Measured length of string = __12.07 cm__or__0.12 m______________ Harmonic, n Tension, T (N) Measured wavelength from ruler ( cm ) Wavelength, λ n (m) Velocity (m/s) 12 14.00 2.13 0.70 70 11 16.66 2.29 0.76 76 10 20.16 2.54 0.84 84 9 24.89 2.77 0.91 91 8 31.50 3.10 1.02 102 7 41.14 3.56 1.17 117 6 56.00 4.14 1.37 137 5 80.64 4.93 1.63 163 μ slope = ___ 3.07 × 10 − 3 ________ kg/m

Table 3 Weight suspended, T = _____50.50_____________ N Frequency, f = _____100.00_____________ Hz Length of string, L = ______4.00_____________ m Measured length of string = __12.07 cm__or__0.12 m______________ Harmonic, n Linear Density, μ (x10 -3 kg/m) Measured wavelength from ruler ( cm ) Wavelength, λ n (m) Velocity (m/s) 2 0.32 11.73 3.87 387 3 0.71 8.18 2.70 270 4 1.26 6.12 2.02 202 5 1.97 4.93 1.63 163 6 2.84 4.11 1.36 136 7 3.87 3.48 1.15 115 8 5.00 3.18 1.05 105 T slope = _______47.43_____________ N

Data Analysis Make sure to show all calculations (or an example of one if repetitive) in an attached calculation page which should follow the questions. Repetitive calculation results should be placed in an appropriate table as directed or as needed. Part I - Wave Velocity Determination 1. Using the measurements that you made of the physical length of the 4.0 m line and the physical distances in Table 1 of the wavelength, calculate the corresponding simulation wavelength. Record this in Table 1. 2. Plot a graph with your data. Frequency should be on the vertical axis and 1/λ on the horizontal axis. Draw a best fit line through your data and determine the slope. The slope should be the wave speed. Record the value in Table 1. 3. Use the weight of the hanging mass, your tension value, and the mass per unit length, μ, to calculate the wave speed, v. Record the value in Table 1. 4. Calculate the percent error between these two values, using the speed you obtained from your slope as the accepted value. Part II – Effect of tension on wavelength 1. Using the measurements that you made of the physical length of the 4.0 m line and the physical distances in Table 2 of the wavelength, calculate the corresponding simulation wavelength. Record this in your data table. 2. Use the wavelengths and the initial frequency to calculate the velocity of the waves and record in Table 2. 3. Plot a graph with velocity on the vertical axis and tension on the horizontal axis. As in previous labs, if the graph is not linear, you MUST determine how to linearize it (consider equation to be verified). Draw a best fit line for data in your linearized graph and determine the slope. Use your slope to determine the mass per unit length, μ. Record the value in Table 2. 4. Calculate the percent error between these two values, using the value from the initial conditions as the accepted value.

The linearization of the graph of my data in Table 2 showed a relationship between the velocity of waves on the string and the tension applied to the string. In order for velocity to directly relate to tension, velocity must be squared. Therefore, velocity squared is directly related to tension. Also, the slope of the graph must be inverse. This is so that it satisfies the equation v= √ T μ . 3. Based on the percent error calculations from your data in Table 2, how well did the mass per unit length gotten from your slope compare to that which you fixed for this part of the experiment? Based on the percent error calculations from my data in Table 2, the mass per unit length gotten from my slope compares to what I fixed for this part of the experiment well. The percent error that I got for this part was 2.54%. This is relatively low; meaning that everything lined up. However, there may be a small error at some point which would cause the percent error that is present. 4. From the linearization of the graph of your data in Table 3, if needed, what is the relationship between the velocity of waves on the string and the mass per unit length of the string? From the linearization of the graph of my data in Table 3, there was a relationship between the velocity of waves on the string and the mass per unit length of the string created. Velocity does not directly relate to Mass per Unit Length; however, Velocity Squared

directly relates to 1/Mass per Unit Length. Therefore, when relating velocity to mass per unit length, velocity must be squared and the inverse of mass per unit length must be taken. 5. Based on the percent error calculations from your data in Table 3, how well did the tension gotten from your slope compare to that which you fixed for this part of the experiment? Based on the percent error calculations from my data in Table 3, the tension gotten from my slope compared to that which I fixed for this part of the experiment well. The percent error that I got was 6.08%. Therefore, the percent error is not that great; so, that means that the data lined up. However, there could be a small error somewhere along the way that would cause this percent error value.