13257 4H14 BEE npiazzacom Oscillator P ' _ r Io _ , . ,. 0Q . L . (11) [7 points] A string with length L and linear mass density [1 is driven by a mechanical harmonic oscillator at frequency f to form standing waves. The tension in the string is provided by hanging mass m over the end of a support. The frequency is increased from zero and standing waves are observed at certain frequencies. The lowest frequency is called the fundamental frequency or first harmonic and is identified with n = 1. The fourth harmonic wave pattern (n = 4) is sketched here. (a) Find a symbolic formula for the nth standing wave frequency. Take the following values: 11 = 4-, m = 0.50 kg, L = 1.6 m, u = 0.002 kg/m, and g = 9.81 m/sz. (b) What is the frequency, speed, and wavelength of the traveling wave that makes up the standing wave pattern. (c) Briefly explain how the traveling wave driven by the oscillator results in the standing wave pattern observed. '5' 29%i: (a) The frequency of the nth standing wave can be found using the formula: f_n= n*v/2L where v is the speed of the wave on the string, which can be calculated using the equation: v= sqrt(T/p) where T is the tension in the string and p is the linear mass density of the string. The tension T can be calculated as: T = mg where m is the mass hanging over the end of the support and g is the acceleration due to gravity. Substituting the given values for n, m, L, u, and g, we have: T = (0.50 kg)(9.81 m/s*2) = 4.905 N v = sqrt((4.905 N)/(0.002 kg/m)) =99.03 m/s f_4 = 4*(99.03 m/s)/(2*(1.6 m)) = 123.8 Hz Therefore, the frequency of the fourth standing wave is 123.8 Hz.(b) The frequency of the traveling wave that makes up the standing wave pattern is the same as the frequency of the nth standing wave, which we found to be 123.8 Hz in part (a). The speed of the traveling wave is given by the equation: V = fA where A is the wavelength of the wave. Rearranging this equation, we have: A=v/f Substituting the given value of v and the frequency of the traveling wave, we have: A = (99.03 m/s)/(123.8 Hz) = 0.798 m Therefore, the wavelength of the traveling wave is 0.798 m.(c) The traveling wave driven by the oscillator reflects back and forth between the two ends of the string. When the reflected wave interferes constructively with the incident wave, a standing wave pattern is formed. The nodes, or points of zero displacement, are formed at the fixed ends of the string, while the antinodes, or points of maximum displacement, are formed at points along the string where the wave undergoes maximum displacement. The standing wave pattern is a result of the interference of the incident and reflected waves, which have the same frequency and wavelength. The frequencies of the standing waves are harmonics of the fundamental frequency, which is the frequency of the first standing wave pattern