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Part A

Standing Waves on a Stretched String Purpose: The purpose of this experiment is to generate transverse waves on a string, and to study the relationship between the tension and the wavelength of a standing wave on a string. You should confirm the fact that the speed of the wave is proportional to the square root of the tension force and the equation v = af. Apparatus: string vibrator, braided line, set of weights, weight hanger, meter stick, sensitive balance, table rod, table clamp, pulley. Theory: The velocity of a transverse wave along a stretched string is given by the equation: V = F VPlinear ( 1 ) Where v is the velocity of the wave, F is tension in the string, and Plinear is the linear density of the string (mass of the string per unit length of the string). If one end of a string is held fixed and the other end is attached to a vibrator, so its direction of vibration is at right angles to the direction of the string, wave disturbances will travel along the string with the velocity, v of Equation 1. At the fixed end the waves will be reflected back along the string. If the tension and the length are adjusted so there is an integral number of half wavelengths in the string, stationary waves will be set up. See Figure 1 for examples of standing wave patterns. Procedure: Set up the apparatus as shown in Fig.2. Connect the vibrator to the oscillator source and set the frequency to 120 Hz. Without adding any weights to the weight holder, pull down on the weight holder with your hand, gradually increasing the tension in the string. If this is done carefully, and if the apparatus is working properly, the string will vibrate in five, four, three, two, and one segments. Sometimes one segment cannot be obtained because the large tension may halt the vibrator. After you have succeeded in making the string vibrate with your hand, repeat by adding weights to the weight hanger. Record the hanging mass (including the mass of the holder) that produces the best set of stationary waves for the case of five, four, three, two and one vibrating segments or "loops". To obtain the highest precision in the above measurements, place weights on the hanger to produce a strong vibration. Be patient and find the best vibration. 91 c- Obtain the wavelength by calculating 2L. This assumes that the vibrator is a node, which introduces a small error. Calculations and Analysis: 1. Measure the linear density of the string by measuring the mass of the string and dividing by its total length (use kg and meters). 2. Using Eq. 1, compute the velocity of the transverse wave in the string for each of the vibrating cases. 3. Notice that the product of frequency and wavelength is equal to the velocity of the wave, calculate the frequency of vibration for each case. 4. Compare the results in 3 with the expected answer. Use the percent error calculation. Actual Value - Experimental Value % error = - x 100 Actual ValueFigure 1 One Segment Two Segments Three Segments Figure 2 A Vibrator C Standing wave D Pulley Weight hanger M S Data: 22 0.91 - Mass of the string = 3-19 = S kg Total Length of the string (used to find density) = 1 80 m Linear density of string (Plinear = mass 0.0319 0100 1696 150 length kg/m NOTE: The length used to calculate the wavelength is the distance between the vibrator and the pulley, but to calculate the linear density you need the total length of the string. Length of the string between the pulley and the vibrator (L) = O.91 m Number of 1 = - 21 Hanging F = Mg "loops" (n) n (N) F f =v/2 Percent Mass V = (Hz) ( Hz ) Error M (kg) Plinear (experiment) (actual) 5 0:364 300 0319 2.94 141.63 117978 1. 86 4 0. USS 3.92 128. 0761 121 -63 3 1. 36 0:607 900 = 09 18-82 1 72 . 1141 89 / 122: 38 2 1-98 0.91 2050 20 -09 108.8371 123 :13 2.61 5. Graph: Plot the square root of the force as a function of the wavelength. Get a straight line-fit to this plot and find the slope of the line. Use Excel, LoggerPro, or another spreadsheet program. Sketch your graph in Figure 3. 6. Calculation: The slope of the line just plotted should be equal to the product of frequency and the square root of the linear density, check that it is the case. 1. 82 = 0.3* 0.98 3 .92 0- 601696 - 2.94Figure 3 J VFIN! V LA 0: 10 01 Giv ( m ) 6 . 6 0 .8 Slope measured graphically (from the graph VF vs. 1): 5 0438 Expected value of the slope (fvp): _. 1bo Now make a prediction, how much mass M would it require to create a single loop (n = 1). Show all your work below. d 26 A $ 20+ 0191 -- 1- 82 323 M (forn = 1) = 7 79 kg If this mass M is too large and dangerous to use in this experiment. How would you modify the experimental set up so that you can produce a single loop with a small mass M. Discuss your modifications with the instructor before changing the experimental set up. What new mass would you need with this modified experimental setup? Show all your calculation below. M (for n = 1) =_ 10 kg (Theoretical) Now perform experiment with the modified setup and find the mass M experimentally that produces one loop. M (forn = 1 ) = 090 kg (Experiment) Calculate the percent error. % error =_ 3- 0%