Consider the uniform string with constant mass per unit length \(m\), constant tension \(T\), and length \(L\) shown in Figure 7.39. One end of the string is attached to a mass \(M\) and the other end of the string is constrained by a spring with stiffness \(k\). Both ends are constrained to move in the vertical direction. Find the frequency equation, the first three natural frequencies, and plot the first three mode shapes. Use the ratios \(m L / M=2\) and \(k L / T=100\).

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777 m, L. T x y(x,t) 00000 Figure 7.39: String mass-constrained at one end and spring-constrained at the other end.