Waves on a String Lab

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University of California, Santa Barbara *

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6BL

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Physics

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Apr 3, 2024

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Lab 3: Waves on a String Physics 6BL 5/6/21 INSTRUCTIONS: - To begin, download a copy of this document to your Google Drive - Go to “File” on the top menu, then click “Make a Copy” from the dropdown menu - Fill in your name, section, TA, and date on the top left corner of this page - You should have 12 pages of questions after this initial page so 13 pages total - You will include this instructions page in your report - Tables, graphs, and questions each have a separate section of the document - Tables and graphs may be copy and pasted or uploaded as an image from the lab specific google spreadsheet - The tables are formatted to fit the information asked for in the lab - All graphs will go on page 4. Please clearly differentiate which graph goes with which exercise - To clarify: You can edit this lab in Google Docs - If you need extra space, please consider changing the size of your work so it fits in the boxes. Contact us if you have trouble with this. - Please ask on Piazza or email us if you have any questions about this format. We are happy to help. - In the end, you should save your report as a pdf and turn it in to the submission portal. Gradescope will not ask you to match pages to questions because you submitted in this format. Your Final Report Pages should be as follows: Page 1 - This Instructions Page Page 2 - Exercise 1 & 2 Data Tables Page 3 - Exercise 3 & 4 Data Tables Tension of string=mass times g Page 4 - All Graphs Page 5 - Questions 1 & 2a/b Page 6 - Questions 2c/3a Page 7 - Question 3b/4a Page 8 - Question 4b/c Page 9 - Question 5/6a Page 10 - Question 6b/7a Page 11 - Question 7b/8 Page 12 - Question 9a/b Page 13 - Question 9c & Conclusion Exercise 1: Constant Values Table

Constants Table System # Linear Density (kg/m) Length (m) Mass (kg) Tension (N) Velocity (m/s) Fundamental Frequency (Hz) System 1 0.003 kg/m 1.70 m 0.4 kg 3.92 36.1 10.6 System 2 0.003 kg/m 1.70 m 0.2 kg 1.96 25.6 7.53 System 3 0.003 kg/m 1.70 m 0.4 kg 3.92 36.1 21.2 Exercise 2: System 1 System 1 Segments Frequency (Hz) 1 10.8 2 21.2 3 31.6 4 43.9 5 54.6 6 64.5 Exercise 3: System 2

System 2 Segments Frequency (Hz) 1 7.1 2 14.0 3 21.4 4 28.3 5 35.6 6 42.8 Exercise 4: System 3 System 3 Segments Frequency (Hz) 1 20.1 2 41.4 3 59.5 4 78.1 5 97.7 6 115.2 GRAPHS/PLOTS (1 or 3 plots)

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- Include all graphs on this page. Make sure to include the exercise number in your title. Example: “Exercise 4: Change in Pendulum Length Affection Period”

Question 1 (partner’s name) The pulse on a denser string will have a lesser propagation velocity than a less dense string. Using the given equation for propagation velocity: velocity= . With string density being the ????𝑖?? ???𝑖?𝑔 ????𝑖?? denominator of the equation, as the string density increases the propagation velocity decreases. Question 2a (partner’s name) velocity=wavelength frequency · wavelength=velocity/frequency wavelength= (5 m/s)/(10 Hz) wavelength= 0.5 m Using Equation 2, wavelength is equal to velocity multiplied by frequency. So the velocity, 5 m/s, multiplied by the frequency 10 Hz, gives us a wavelength of 0.5. Question 2b wavelength=velocity/frequency wavelength=(20 m/s)(10 Hz) wavelength=2.0 m If the velocity was 20 m/s, the wavelength would be 2.0 m.

Question 2c The sinusoidal pattern on the string is longer with greater propagation velocity. Question 3a λ=2L/n λ=2(0.75 m)/(5 wave segments) λ=0.30 m A wavelength of 0.30 m is required to get 5 standing wave segments on a doubly fixed string of length 0.75 m.

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Question 3b frequency=velocity/wavelength frequency=(50 m/s)/(0.3 m) frequency=170 Hz The frequency will be 170 Hz when the velocity is 50 m/s. Question 4a Tension=T=mg T=(0.4 kg)(9.8 m/s 2 ) T=3.92 N The tension in the string is 3.92 N.

Question 4b velocity=(tension/density) (½) velocity=(3.92 N/0.003 kg/m) (½) velocity=36.1 m/s 2 The speed of the wave is 36.1 m/s 2 . Question 4c f n =(1/2L)( ) 𝑇 µ v= 𝑇 µ f n =(1/2L)(v) f n =(½)(1.7 m)(36.1 m/s 2 ) f n =10.6 Hz The fundamental frequency of the system is 10.6 Hz.

Question 5a Ratio: Fundamental of System 2/Fundamental of System 1 (7.53 Hz)/(10.6 Hz)= 0.71 Hz Question 5b Halving the mass also leads to halving the tension. The fundamental frequency depends on the root of the tension, so the factor of change from a half mass should be 0.71. Question 6a Ratio for System 3/System 1 = 2:1 (21.2 Hz)/(10.6 Hz)= 2

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Question 6b Halving the length, would change the frequency by a factor of (1/2(0.5)). This comes from the f 1 =(1/(2L)). The factor should be (1/2(0.5))/(1/2(1.0))=2. Question 7a System f 2 /f 1 f 3 /f 1 f 4 /f 1 f 5 /f 1 f 6 /f 1 1 1.96 2.93 4.06 5.06 5.97 2 1.97 3.01 3.99 5.01 6.03 3 2.06 2.96 3.89 4.86 5.73 Sample Calculation for System 1 f 2 /f 1 : f 2 /f 1 = 21.2 Hz/10.8 Hz= 1.96

Question 7b Halving the length or the mass did not affect the relationship between the upper harmonics and the fundamental. The ratio was fairly constant. The ratio being f n /f 1 . Question 8 The slope for System 1 is 10.9. The slope for System 2 is 7.15. The slope for System 3 is 18.9.

Question 9a/Exercise 5 work Using the Equation 5, the slope of the graph is f 1 . Where f 1 =(1/(2L))/(tension/density) (½) . Question 9b The units for slope are in Hz. The physical meaning is that each slope is the first harmonic for each given system.

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Question 9c System Observed f 1 (Hz) f 1 from slope (Hz) Percent difference 1 10.8 Hz 10.9 Hz 0.92% 2 7.1 Hz 7.15 Hz 0.70% 3 20.1 Hz 18.9 Hz 6.3% Percent Difference= 100% 𝐸???????−𝐸????𝑖????𝑎? 𝐸??????? | | | | × Sample Calculation for System 1: 100 = 0.92% 10.9−10.8 10.9 | | | | × CONCLUSION In this lab, we observed different factors that could affect the speed of a wave on a string. We also tested the pulse segments of a string at different frequencies. The observed frequencies were noted and then graphed to get an expected value for frequency for each system. Overall, the understanding was the relationship between frequencies, velocity, tension, and wavelength.

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