Experiment 1: Standing Waves in a String The purpose of this lab is to investigate standing waves in a string and to check the relationships between the frequencies of the standing waves, the tension in the string, the string linear mass density, and the oscillation patterns of different modes. Theory Standing waves of many different wavelengths A, , n =1, 2,... can be produced on a string with two fixed ends, as long as an integral number of half wavelengths fits into the length of the string. For a standing wave on a string of length [ with two fixed ends AT = 21 n = 1, 2,3, . . . (1) Several oscillation patterns are shown on the Figure. Fundamental 1 st Harmonic Fundamental: n=1; one half wavelength fits into the length of the string First Overtone 2nd Harmonic . Second harmonic: n=2; two half wavelengths fit into the length of the string Second Overtone ard Harmonic Third harmonic: n=3; three half wavelengths fit into the length of the string Third Overtone 4th Harmonic And so on..The integer number n corresponds to the number of the half-wavelengths, or segments, fitting into the length of the string. The speed of a transverse wave in a string is given by (2) where T' is the tension in the string measured in Newtons and / is the linear mass density of the string measured in kg/m . Using Egs. (1) and (2) in the wave relation o = Af one can derive the following expression for the frequencies f, of the standing waves T1 fn = 4 2L n , n = 1, 2,3, ... (3) In this lab, we'll use two different sets of measurements to explore experimentally Eq. (3) Physics Simulations: Wave on Strings Lab 1 http:://www.geogebra.org/m/vv2xwywh Experiment: For all runs set linear mass density to A = 1 x 10 * kg/mExperiment: For all runs set linear mass density to / = 1 1 10 * kg/m Part A: Varying Tension at Constant Driving Frequency In this experiment, we'll keep the driving frequency f constant and gradually change the tension T in the string until one of the standing wave frequencies fo in Eq. (3) coincides with the driving frequency, and the standing wave with n segments is excited due to the resonance. Keep increasing tension T and eventually fa-1 will coincide with driving frequency f and the standing wave with n - 1 segments is excited, then fn 2, ... etc. It follows from Eq. (3) that tension T' corresponding to the excitation of the standing wave with n segments at given driving frequency f is T = (412 12 14 ) (212 ) (4) Measurements 1. Set the driving frequency f =100 Hz, and set the tension T' =10N. (These values of parameters should give you the standing wave with n =8 segments) 2. Start increasing the tension until you excite the standing wave with n =7 segments and record your result in the data table. (Help: you can change the tension roughly by sliding the radio button on the experimental panel and make fine tuning by clicking on the right or left side of the button) 3. Continue to increase the tension until you find all standing waves with number of segments ranging from n =8 to n =1 and record your result in the data table. (Hint: it's a good idea to estimate the required tension roughly using Eq. (4)) Driving frequency f: 100 HzString length L: 4m Segments, n Tension, T (Newtons) 1 12 2 8 0.0156 7 0.0204 6 0.0278 un 0.0400 0.0625 0.1111 2 0.2500 1 1 Analysis If the tension is varied while the length and frequency are held constant, a plot of Tus (1 ) should be a straight line with a slope equal to 41/3f