All Problems Mechanical Oscillations
Problem 4.69
A point performs damped oscillations with frequency \(\omega\) and damping coefficient \(\beta\) according to the law \((4.1 \mathrm{~b}) .\) Find the initial amplitude \(a_{0}\) and the initial phase \(\alpha\) if at the mòment \(t=0\) the displacement of the point and its velocity projection are equal to (a) \(x(0)=0\) and \(v_ {x}(0)=\dot{x}_ {0}\) (b) \(x(0)=x_ {0}\) and \(v_ {x}(0)=0\)
4.69. (a) \(a_{0}=\frac{\left|\dot{x}_{0}\right|}{\omega}, \quad \alpha=\left\{\begin{array}{ll}-\pi / 2, & \text { when } & \dot{x}_{0}>0, \\ +\pi / 2, & \text { when } & \dot{x}_{0}<0 ;\end{array}\right.\) (b) \(a_{0}=\) \(=\left|x_{0}\right| \sqrt{1+(\beta / \omega)^{2}}, \quad \alpha=\arctan (-\beta / \omega),\) with \(-\pi / 2<\alpha<0,\) if \(x_{0}>0\) and \(\pi / 2<\alpha<\pi,\) if \(x_{0}<0\)