A bead is released from the position x(0) x0 on the curve z 1 2x2 with initial velocity
Chapter 11, Problem 5(choose chapter or problem)
The Sliding Bead
A bead is released from the position \(x(0)=x_{0}\) on the curve \(z=\frac{1}{2} x^{2}\) with initial velocity \(x^{\prime}(0)=v_{0} \mathrm{~cm} / \mathrm{s}\).
(a) Use the phase-plane method to show that the resulting solution is periodic when the system is undamped.
(b) Show that the maximum height \(z_{\max }\) to which the bead rises is given by
\(z_{\max }=\frac{1}{2}\left[e^{v_{\alpha}^{2} / g}\left(1+x_{0}^{2}\right)-1\right]\)
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