Experiment 303: Transverse Wave: Frequency of Vibration Raagas, Michelle Mae G. School of Chemical Engineering, Chemistry, Biological Engineering, and Material Science Engineering Mapua Institute of Technology, 658 Muralla St., Intramuros, Manila City, Philippines [email protected]
OBJECTIVE: The purpose of this experiment is to analyze the concept of frequency of vibration with respect to the transverse wave. To verify each stretched string’s frequency of vibration is the main goal of this experiment. In addition, it also aims to identify how the tension and linear mass density affects the vibrating string’s frequency.
On this experiment, each group is given a specific frequency and the assigned frequency on our group is 96.30 Hz. First thing to do after the set-up of the apparatus is to adjust the amplitude and frequency depending on the assigned frequency given by the professor. It can modify using the two knobs located at the upper right of the sine wave generator. See figure 2.
METHODOLOGY: For the objective to be accomplished. The use of equipment is necessary for this experiment. It will help us to collect different sets of data which will be useful to achieve the objectives. Below are the materials used in this experiment.
Figure 2. Sine wave generator
Figure 1. Materials used for the experiment “transverse wave: frequency of vibration” (string vibrator, sine wave generator, iron stand with clamp, pulley, weights, mass hanger, extension cord, meter stick, and guitar string).
The goal of the part 1 of this experiment is to determine the frequency of vibration with constant linear mass density which means that the diameter of the wire is constant and also its linear mass density. To do so, choose the size of the string to be used for this entire part of the experiment, carefully tie the end of the guitar string on the string vibrator and the other end is attached with a mass hanger and is then hanged over a pulley. The next step is to add a certain mass on the mass hanger. Observe the string until it forms
a segment. You can adjust it using the amplitude knob or by adjusting the distance between the two iron stand.
Repeat the said step for five times but this time, increase the mass that is added on the mass hanger every trial. Notice if the string has a curve edges because it can cause errors and more difficult to form a standing wave. See the figure below.
Figure 3. Adjusting the distance and amplitude until the string forms a standing waves When you see a standing wave (refer to figure 4) measure its length and record it also the number of segment that is formed.
Figure 5. Curved edges After getting all the data, you can now solve the frequency of vibration that has a constant linear mass density. The process on the part two of the experiment is the same as the part one but unlike in the part one that you will only use one string, here in part two, you will use five different strings with different diameter and linear mass density. You will use the same mass on the mass hanger for every trial. The linear mass density of the string is already given depending on the diameter of the string that is used. After you gather all the data that is needed, you are now able to solve the frequency of vibration with variable linear mass density. Figure 4. Standing waves
DATA and SAMPLE COMPUTATIONS Table 1. determining the frequency of vibration (constant linear mass density) Diameter of wire: 0.022 in. Linear Mass Density= 0.0184
Trial Tension, T
(mass of pan) x 980 cm/𝑠 2
Frequency of vibration
1 24500 dynes 1 6 2 29400 dynes 1 7 3 34300 dynes 1 8.5 4 39200 dynes 1 9 5 44100 dynes 1 9 Average frequency of vibration Actual value Percentage error
𝑛 𝑇 √ 2𝐿 𝜇
19.16 Hz 90.29 Hz 80.31 Hz 81.09 Hz 86.01 Hz 86.77 Hz 96.30 Hz 9.89 %
86.77 − 96.16 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑒𝑟𝑟𝑜𝑟 = | | = 9.89 % 96.16 Table 2: 𝑛
Trial 1: 𝑓 = 2𝐿 √𝜇 = 2(18) √0.0184 = 88.07 𝐻𝑧 Average frequency =
= 89.18 Hz 𝛼𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 − 𝛼𝑎𝑐𝑡𝑢𝑎𝑙 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑒𝑟𝑟𝑜𝑟 = | | 𝛼𝑎𝑐𝑡𝑢𝑎𝑙 89.18 − 96.16 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑒𝑟𝑟𝑜𝑟 = | | = 7.4 % 96.16
GRAPH Table 2. Determining the frequency of vibration (variable linear mass density) n L 1 1 1 1 1
18 12 10 9 9
f Hz 88.07 93.41 93.54 89.81 81.09 89.18 96.30 7.4%
T 𝜇 dynes g/cm 1 0.010 in 0.0039 39200 2 0.014 in 0.0078 39200 3 0.017 in 0.0112 39200 4 0.020 in 0.0150 39200 5 0.022 in 0.0184 39200 Average frequency of vibration Actual value Percentage error Sample computation Trial diameter
Tension vs. Frequency of vibration
Trial 1: 𝑓 = 2𝐿 √𝜇 = 2(6) √0.0184 = 96.16 𝐻𝑧 Average frequency =
= 86.77 Hz 𝛼𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 − 𝛼𝑎𝑐𝑡𝑢𝑎𝑙 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑒𝑟𝑟𝑜𝑟 = | | 𝛼𝑎𝑐𝑡𝑢𝑎𝑙
20000 0 86.01
Frequency of vibration
Graph 1. Frequency of vibration with constant linear mass density
Linear mass density
Actual value of frequency of vibration: 96.30 Hz Table 1:
Linear mass density vs. Frequency of vibration 0.0184
Frequency (Hz) Frequency of vibration
Graph 2. Frequency of vibration with variable linear mass density.
ANALYSIS OF DATA
On the part 1 of the experiment, we used the string that is 0.022 inches in diameter since one of my groupmates suggested it.
I would like to thank everyone for helping me in my studies especially God for guiding me. Also, I would like to thank my group mates for the participation and for making this experiment fun and interesting just like when the string is starting to form standing wave we were like ‘whoa great’. I would also like to thank our professor Mr. Ricardo De Leon for teaching us the process in this experiment and for being my professor also on my physics lecture that helps me to understand the concept behind the experiment. I would also like to recognize the effort of the lab assistant who provides us the experiment and for explaining the proper way to handle it. Lastly, I would like to thank my friend for making me a cup of coffee while I was doing this lab report. Thank you so much!
Based on the gathered data, since the diameter and the linear mass density of the wire is constant, we observed that as we add mass to the pan, the tension also increases as well as the length of string that forms segment of wave. It is because we adjusted the distance between the two iron stand until it forms wave. From the time that we change the tension of the string, the number of segment and the length of it will also change. On the second part of the experiment, the mass on the pan for each trial and so is the tension. This time, the one that is changing is the diameter of the wire and the linear mass density, and we also observed that as the diameter of the wire increases, the length of the string’s standing wave decreases. There are factors that can affect the experiment and can cause earlier. One of these are the strings that are used. As I said earlier, it is more difficult for the string to form standing waves if it has some curve edges.
CONCLUSION By performing this experiment, the two objectives of the experiment were obtained by applying the concept of transverse wave and relating it on our experimentation. By using different kind of string, we are able to determine the string’s frequency of vibration. The second objective of this experiment which is to determine how the linear mass density and the tension can affect the vibrating string’s frequency was attained by performing the two parts of the experiment. The first one is the effect of tension with constant linear mass density and the second is the reverse of it which is the tension is constant and the linear mass density is the one that is changing. In addition, I therefore conclude that 1 𝑓 ∝ √𝑇 𝑎𝑛𝑑 𝑓 ∝ 𝜇. √
REFERENCES  Halliday, F., Fundamentals Of Physics, 9th Edition, 2011  King, G., Vibrations and waves, p.137