In moderate winds, high-tension power lines sometimes “sing” or hum as a result of trans...

In moderate winds, high-tension power lines sometimes “sing” or hum as a result of transverse waves on the lines that winds can generate. The cables that carry high-voltage electricity are made of aluminum with a steel core added for strength. The cross sectional areas of the steel and aluminum in the line are and respectively. The cable is 1.8 cm in diameter and is under a tension of Typically, the towers are placed about 0.20 km apart. (See Figure P12.71.) (a) What is the fundamental frequency of such a power line? (The densities of steel and aluminum are given in Table 10.1.) (b) The wind can induce vibrations when flowing around a cylinder. The frequency of the vibrations induced this way is modeled well by the equation where is the speed of the wind and d is the diameter of the cylinder. The wind will induce vibration this way only if the frequency of the induced oscillation matches a natural frequency of the power line. It is found that the power line hums when the wind blows at a steady 6.7 m/s. What harmonic of the line is this induced vibration exciting? (c) If the line is humming at the 152nd harmonic, how fast is the wind blowing?

(reference table 10.1)