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2 Experiment P09: Waves Waves Experiment P09 Objective Study the formation of standing waves on a vibrating string Materials Digital Function Generator Digital Scale String Vibrator Mass Hanger, Large, 50 g Large C-Clamp Banana-Banana Cable, x2 (1 red, 1 black) Elastic Wave String Slotted Mass Set Pulley w/Table Clamp Meter Stick Power Cable Theory A standing wave is one that oscillates with time but remains fixed in its location. In a string of length L that is tied down at both ends, standing waves are visible on the string if the string is vibrated. The standing waves have a sinusoidal appearance-equal displacement above and below a zero-amplitude line. The points on a standing wave that remain fixed (zero displacement) are called nodes, and the points of maximum displacement are called anti-nodes. The frequency of a standing wave is equal to the number of complete waves that pass a fixed reference point in one second. The wavelength of a standing wave is equal to the distance between consecutive anti-nodes. The velocity of a wave traveling along a string depends only on the properties of the string, not on the frequency or wavelength of the wave The velocity of a wave traveling on a string is given by: Tension mass per unit length VE (1) where v is velocity, Fis the weight (in Newtons) of the mass m hanging from the string, and o (sigma) is the linear density of the string in units of kilograms/meter. The velocity v, frequency f, and wavelength ) of a wave are related by: v = fx (2) For standing waves on a string, the wavelength, A is twice the distance between nodes () = 2d). However, because the period T and frequency fare related by: T = - (3) we can express also wavelength A as: A = VT (4){lid Dominion University Physics 111E26i231i251 Lab Manual 3 The linear density [mass per unit length} of the string is found by: o =1 [s] I. where M is the mass of the string in ldloggarn, and L is the length of the string in meter. Procedure in this experiment, you will use a frequency generator [Pasco Digital Function Generator] to drive a speaker that vibrates one end of a string. The other end of the string passes over a pulley and has a slotted mass hanging from it to provide tension in the string The tension [I41 in Equation {1} is the weight [in Newton] of the slotted mass {m}, hanging from the string The linear density {or} is in kilogramfmeter ['icgfm]. For standing waves on a string. the wavelength [1].. is twice the distance between nodes [1 = 2d]. 1 Equipment Setup 1. Measure the length of the elastic wave string. (Note: all loiots must be removedl} Use a di'ta] scale to measure the mass of the elastic wave string. Record the values in Data Table 2. 2. Take the large Cshaped clamp and use it to secure the String Vibrator to the table. There is even a spot on the vibrator that says, \"Clamp Here". 3. Plug one red and one black banana-banana cable into the redfblack ports on the Function Generator and then plug the other end into the string vibrator. 4. Attach the Pulley table clamp to the opposite edge of the lab table. Figure H.111: Shim Vibrator Figure P012.- Pulley 5. a] Tie a small loop on the end of the elastic wave string hanging over the pulley assembly. b] Attach a 5D g mass 4 Experiment PDQ: Waves hanger to the elastic wave string han' ng over the pulley. Then add 45D grams to the hanger. 6. a] Align the position of the string vibrator and pulley assembly so that the elastic wave string forms a straight line when viewed from above. b] Adjust the height gigging}: so that the elastic wave string forms a horizontal line with respect to the table. Data Collection Turn-on the function generator and adjust the frequency knob to reduce the frequency to between 2-10 Hz. Set the function generator to produce a sinusoidal [N] waveform [the top choice}. ['1 2. Adjust the frequency to nd the rst harmonic standing wave in the string [you will see one antinode with a node on each end]. Finely tune the 'frequency' knob as necessary until you get the largest antino de possible. The amplitude knob should probably be somewhere in the middle of its range. 3. Measure the distance between nodes for the rst harmonic standing wave and note the frequency Record these values in Data Table 1. 4-. Slowly increase the function generator frequency until the string vibrates at the second harmonic standing wave [2 antinodes. 3 nodes]. Make slight adjustments to the frequency and amplitude controls to obtain the maximum amplitude of antinodes. Measure the distance between all adjacent nodes and record the frequency. Record the values in Data Table 1. 5. Find as many consecutive harmonics [# ofantinodes] as you can see on the string until you have reached the tenth harmonic [ll] antinodes]. Measure the distance between all adjacent nodes and note the frequency for each harmonic found. Record the values in Data Table 1. 6. For each harmonic. calculate the average distanoe between the nodes and then calculate the wavelength of the standing wave. Record the values in Data Table 1. Show your work for the third harmonic. T. For each harmonic, use Equation [2} to calculate the velocity of each standing wave. Record the values in Data Table 1. Show your work for the third harmonic in the data section of your lab report. For the linear density of the string use the value o = D. [10295 kgy'm Because our string is elastic, and stretches as we add tension, the linear density is not constant. We have obtained this average value from previous semesters and should work well for your experiment. 8. Find the average of all your wave velocities and record this in Data Table 2 as um. 9. Use F=ma to calculate the tension in the string [don't forget to include the mass of the hanger along with any rnass you added}; and then use Equation [1] to calculate the theoretical velocity Old Dominion University Physics 111l226l231l261 Lab Manual 5 of waves on the string. Record the value in Data Table 2 as othemcal. Show this calculation in your lab report. 1D. Calculate the percent error between the velocity of the waves on the string from the graph and the theoretical velocity calculated from Equation [1]. Record the value in Data Table 2. Show your work in the data section of your lab report. theoretical velocityaverage velo citj.r X 100 D : XE] ETTOT theoretical velocityr Graphing 1 vs T 11. In a spreadsheet program of your choice, create two columns: one column with your frequency data and one with your wavelength data. 12. In a third column, have your program calculate the period Tof the wave at each frequency. This comes from equation [3]. 13. Create a plot of wavelength versus the period. Be sure to correctly label the axes of the graph and include units. 14. Add a linear trendline to the data and display the equation on the chart 15. Use the slope of the cloth, in conjunction with Equation [4], to determine the velocity with which the waves travel along the string. How does this value for the wave velocity compare to the value you found before and the theoretical value? Data Table - Experiment P09 Data Table 1 Standing Waves on a String Wavelengt Harmonic Average Frequency h of Velocity of (# Distance of Harmonic Period Harmonic the Wave 2 = 2d v = fx T = 1/f Anti-nodes) Between H Nodes (m) (s) (Hz) (m/s) 1 2.155 8 4.310 34.48 0.125 2 1.078 17 2.156 36.652 0.059 3 0.718 25 1.436 35.900 0.040 4 0.539 33 1.078 35.574 0.030 5 0.431 41 0.862 35.342 0.024 6 0.359 50 0.718 35.900 0.020 7 0.308 57 0.616 35.112 0.018 8 0.269 66 0.538 35.508 0.015 9 0.239 74 0.478 35.372 0.014 10 0.216 83 0.432 35.856 0.012Wavelength (m) v. T (s) 34.6*x + 0.0271 Wavelength (m) N 0 0.025 0.050 0.075 0.100 0.125 Period (s) Data Table 2 Standing Waves on a String - Velocity Comparison Linear Density of String (o) 0.00295 kg/m Mass of Hanging Object (m) 0.500 kg Tension on the String (F) 4.9 N Average Wave Velocity (Uave) 34.6 m/s Theoretical Wave Velocity (Utheoretical) 40.76 m/s % Difference (ave VS Utheoretical) 15.11 %