Lab02_StandingWave_ConnorEdwards

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Apr 3, 2024

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Standing Wave on a String Connor Edwards PHY-152 Abstract The motion of a standing wave is one of almost an illusionary nature due to how it behaves to the naked eye. At the right frequency, a standing wave looks superimposed over itself as it vibrates back and forth, especially when dealing with string motion. Therefore, it is not only scientifically interesting but also visually appealing to those with no knowledge of the subject. In this experiment, we perform that exact motion with a string to verify whether the given equations are valid representations or not. To do this we attach a hanging weight to the string at one end and oscillate it with a generator at the other to create a standing wave. By increasing the weight and adjusting the generator’s frequency we are able to record the data and calculate our own gravitational constant being 9.13 m/s^2 and compare it to the known constant of 9.81 m/s^2 to evaluate the differences and sources of error in the design. Introduction Standing waves, also known as stationary waves, are the result of two waves moving in opposite directions with equal frequency and amplitude. They effectively cancel each other out due to their opposing forces, which causes them to create a fixed area of oscillating waves. The interference created equates to effectively zero energy being created by the system, at least for a string. Also, the resultant amplitude will be equal to the amplitude of the individual forces. Therefore, the product will be one like the experiment shown here no matter the specific energy put into the system. Experimental Setup To properly begin this lab, one must first get all the materials together including the high function generator which drives the entire lab by providing the oscillation of the string. Then, measure, weigh, and find the density of a 2-meter-long string to record data (fig. 1) by attaching it to the generator and adding mass while slowly increasing the frequency until observing the number of loops. Repeat this step similarly while adding more mass by 50 grams each trial and record in a new data table (fig. 2), converting to kilograms. Process the data into an excel table, and then calculate the percent error for the calculated gravity and the given gravity. Fixed Data Ls= c m Total string length L= c m String length between pulley and wave driver ms = g Total string mass mh = 5 0 g Mass of mass hanger Fig. 1 Trial m [kg] f [Hz] f [Hz] ^2 1 250 --- --- 2 300 --- --- ~~~ ~~~ ~~~ ~~~ 10 700 --- --- Fig. 2
The picture shows the ideal wave motion for the lab, being the third harmonic at a stable frequency to create 3 oscillating sections of string. This picture shows what would be seen on the frequency generator during the in-person lab demonstration as the value for frequency for the third harmonic in Hz. Experimental Data Fixed Data Ls= 225 c Total string length
m L= 160. 5 c m String length between pulley and wave driver ms = 370 g Total string mass mh = 50 g Mass of mass hanger Table used from Fig. 4 of the handout to record the values measured with a meter stick and digital scale in the second demonstration video. Trial m [g] f [Hz] 1 250 36.1 2 300 39.7 3 350 43 4 400 46 5 450 49 6 500 51.6 7 550 54.1 8 600 56.7 9 650 58.9 10 700 61.3 Table for raw measurements before any conversions or calculations that were taken during the testing trials in the third and fourth demonstration videos. Analysis Calculations: -μ = m/L so 370g/225cm or .37kg/2.25m = 0.164 kg/m -f & f^2 = f(3)/3 & f(3) squared respectively so, Trial 1 is 36.1/3 = 12.03 or 12 & 36.1 squared is 1303.2 Repeat for Trials 1-10 -f^2=(g/4*L^2*μ) *m reorganize to find g= (f^2/m)*4L^2*μ This gave the answer 9.13 m/s^2. -Percent error = (measured-accepted/accepted) x100 so, (9.81-9.13/9.81) x100 = 7% error
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0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 500 1000 1500 2000 2500 3000 3500 4000 f(x) = 5438.27 x − 55.53 R² = 1 Mass v. Frequency Hanging Mas (kg) Fundamental Frequency (Hz^2) Step 19 Graph Trial m [kg] f [Hz] f [Hz] ^2 1 0.25 12 1303.2 2 0.3 13.2 1576.1 3 0.35 14.3 1849 4 0.4 15.3 2116 5 0.45 16.3 2401 6 0.5 17.2 2662.6 7 0.55 18 2926.8 8 0.6 18.9 3214.9 9 0.65 19.6 3469.2 10 0.7 20.4 3757.7 Data Table for Fig. 6 on handout Results As stated/shown in the analysis section, the calculated gravitational constant came out to be 9.13 m/s^2 with a percentage error of roughly 7%. This means that it was wrong by about 7 percent which is relatively large for performing calculations. Causes for this could be anything from faulty equipment to misreading measurements or values taken from the experiment. The most likely cause is simple human error that would stem from the frequency recorded as being the most stable one, since it is very easy to make mistakes here. So, with all respect to the demonstration videos, with more precise testing equipment the constant found would probably be much closer to 9.81 m/s^2. Conclusion As shown by the data and explanation of the procedures used, the equation given for the lab can be verified. This is because although the percentage error for the calculated value was relatively high, we were still able to show the correct procedure and get an answer even if it was not exactly correct. Also, the videos provided were able to easily demonstrate the procedure and help the student arrive at their own conclusions when needed.
References Lab Videos/Handout Standing wave | Definition & Facts | Britannica 14.7: Standing waves - Physics LibreTexts Additional Questions I am slightly confused on this section since the answers are given and I am unsure of exactly what ‘part b’ is referring to in the section below. If you would like me to answer and resubmit I would be happy to but I don’t get what to do and got home late and forgot until now to submit/email you about it sooner. This section will be required occasionally, as here in Lab 2. In this case, remember that diagrams and/or explanations will be needed for a perfect score. Strive to get the answers in part b.