Physics lab

Lab: Standing Waves Purpose: If a string is fixed at one end and attached to a sinusoidally driven oscillator at the other end, waves travel down the string, reflect, and move back towards the fixed end. Under the correct conditions, standing waves are formed from the interference of the waves traveling back and forth along the string. During this laboratory exercise, the student will use a standing wave simulation and determine values of string tension that cause the standing waves to clearly appear. In order to check the outcome of the experiment, the student will experimentally derive the frequency of the waves. Equipment: (on-campus) scissors, string, mass holders with slotted masses, meterstick, clamps, rods, pulley, 60-Hz AC string oscillator (at home) internet access, phone/tablet/computer Theory: 𝜆= 2𝐿 𝑁 (equation 1) 𝑣 = 𝑓𝜆 𝑣=√ 𝑇 𝜇 1 𝑇 𝜆 = 𝑓 √𝜇 (equation 2) Procedure: ON-CAMPUS 1. Clamp the oscillator and the pulley on opposite sides of the laboratory table. You will also need a piece of string long enough to reach from the oscillator to the pulley and also hang over the pulley as shown in the figure below: Figure: Apparatus setup 2. The mass per unit length  of the string must be determined. Measure the total length of string and measure its mass using a laboratory balance. Record the values of length and mass in the following Data Table – String. Calculate the mass per unit length from your data and record this in Data Table – String. Data Table – String Length Ls (m) Mass ms (kg) Linear mass density = ms/Ls =  (kg/m) 3. Tie one end of the string onto the oscillator and make a loop at the end of the string that hangs over the pulley. Place a small mass holder in this loop. 4. Carefully measure the length of the string from the tip of the oscillator to the point at the very top of the pulley where the string touches the pulley, as shown in the Apparatus Setup figure. Record this distance as L in the Data Table – Standing Waves. 5. Plug in the oscillator and place on the mass holder whatever mass is necessary that produces a standing wave which corresponds to the N = 3 case, shown in the Theory section of this laboratory script. To determine the mass necessary to produce this specific standing wave, it might be useful to pull slightly down or light slightly up on the mass hanger, so you know whether you need to add or subtract mass. Determine the amount of mass M needed to produce the largest amplitude of vibration with N = 3 and record this value in Data Table – Standing Waves. Be sure to include the mass of the mass holder in this total M. 6. Remove mass from the mass holder until a standing wave with N = 4 is achieved. Determine the mass needed to produce the largest amplitude of vibration with N = 4 and record this value of mass M in Data Table – Standing Waves. 7. Repeat this procedure for each of the cases N = 5, 6, 7, 8, 9. Note: N also refers to the number of segments into which the string length is divided, if that helps to make it easier to decide which N value you are seeing. Record the mass M in Data Table – Standing Waves. Data Table – Standing Waves L = _____________________ (m) N Hanging mass M (kg) 3 4 5 6 7 8 9 8. For each value of N, calculate the tension T of the string by taking the mass M and multiplying it by g, the acceleration due to gravity. Record these values of tension T in Calculations Table – Standing Waves. 9. Calculate the values of the square root of the tension √𝑇 and record them in Calculations Table – Standing Waves. 10. Using the equation 1 from the Theory section that relates length L, N, and wavelength , calculate the wavelength  for each corresponding value of N, and record these values in Calculations Table – Standing Waves. 11. Make a graph with wavelength  on the y-axis and the square root of the tension √𝑇 on the x-axis. 12. Plot a best-fit line to your data and record the slope in Calculations Table – Standing Waves. 13. According to equation 2 in the Theory section, the slope of your graph should be equal to 𝑓 1 𝑒𝑥𝑝 √𝜇 . Using your known value of , solve the equation for the experimental value of the frequency fexp and record this value in Calculations Table – Standing Waves. Calculations Table – Standing Waves N Tension T (N) √𝑇 (√𝑁) Wavelength  (m) 3 4 5 6 7 8 9 Slope = Experimental frequency fexp = AT-HOME 14. Go to: https://ophysics.com/w8.html . The screen should look like this: 15. Before making any measurements, let’s play with the simulation first. What do you notice as you change the vibration frequency, leaving everything else the same? 16. What do you notice as you change the linear density of the string, leaving everything else the same? 17. What do you notice as you change the tension, leaving everything else the same? 18. Now hit the Reset button. Record the following values here: Vibration frequency f: _______________ Hz Linear density : ______________ kg/m Length of the string L: _____________ m 19. Determine the amount of tension T needed to produce the largest amplitude of vibration with given N, and record this value in Table – Standing Waves. Remember: N = node = location that does not appear to be moving. (See image in Theory section). N also refers to the number of segments into which the string length is divided, if that helps to make it easier to decide which N value you are seeing. 20. Repeat this procedure for each of the cases N = 5, 6, 7, 8, 9,… given in the table Record the tensions in the Table. You might not be able find all values of N, but try to get as many as you can! 21. Calculate the values of the square root of the tension √𝑇 and record them in Table – Standing Waves. 22. Using the equation 1 from the Theory section that relates length L, N, and wavelength , calculate the wavelength  for each corresponding value of N, and record these values in Table – Standing Waves. Table – Standing Waves N Tension T (N) √𝑇 (√𝑁) Wavelength  (m) 4 5 6 7 8 9 10 11 12 23. Make a graph with wavelength  on the y-axis and the square root of the tension √𝑇 on the x-axis. 24. Plot a best-fit line to your data and calculate the slope. According to equation 2 in the Theory section, the slope of your graph should be equal to 𝑓 1 𝑒𝑥𝑝 √𝜇 . Using your known value of , solve the equation for the experimental value of the frequency fexp and record this value along with the slope below. (Be prepared to upload graph with slope.) Slope = ________________ Experimental value for frequency fexp = _______________ Hz EVERYONE – be prepared to upload this page on the quiz, if required. Error Analysis: Using the known value for the frequency, (on campus = 60 Hz, at home = 125 Hz), calculate the percentage error for your experimental value of the frequency, fexp. Questions: 1. How is the length of the string L related to the wavelength  for standing waves? 2. What is the longest possible wavelength  for a standing wave in terms of the string length L? 3. Calculate the velocity for the N = 2 and N = 8 standing waves. 4. Calculate the tension T that would be required to produce the N = 1 standing wave for your string. 5. In this lab, a piece of string oscillates horizontally with the tension coming from masses hanging on the string as it passes over a pulley. What would be the effect if the string stretched significantly as the tension increased? How would that have affected the data?

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