Solved: When the magnitude of tension T is not constant, then a model for the deflection

Chapter 3, Problem 28

(choose chapter or problem)

When the magnitude of tension T is not constant, then a model for the deflection curve or shape y(x) assumed by a rotating string is given by

\(\frac{d}{d x}\left[T(x) \frac{d y}{d x}\right]+\rho \omega^{2} y=0\) .

Suppose that 1<x<e and that \(T(x)=x^{2}\).

(a) If  y(1)=0, y(e)=0, and \(\rho \omega^{2}>0.25\), show that the critical speeds of angular rotation are

\(\omega_{n}=\frac{1}{2} \sqrt{\left(4 n^{2} \pi^{2}+1\right) / \rho}\)

and the corresponding deflections are

\(y_{n}(x)=c_{2} x^{-1 / 2} \sin (n \pi \ln x)\), \(n=1,2,3, \ldots\)

(b) Use a graphing utility to graph the deflection curves on the interval [1, e] for n=1,2,3. Choose \(c_{2}=1\).