Serway & Jewett — Wave Motion: Problem 55

With particular experimental methods, it is possible to produce and observe in a long, thin rod both a transverse wave whose speed depends primarily on tension in the rod and a longitudinal wave whose speed is determined by Young’s modulus and the density of the material according to the expression \( v = \sqrt{Y/\rho} \). The transverse wave can be modeled as a wave in a stretched string. A particular metal rod is 150 cm long and has a radius of 0.200 cm and a mass of 50.9 g. Young’s modulus for the material is \( 6.80 \times 10^{10} \text{ N/m}^2 \). What must the tension in the rod be if the ratio of the speed of longitudinal waves to the speed of transverse waves is 8.00?