All Problems Mechanical Oscillations
Problem 4.68
A body performs torsional oscillations according to the law \(\varphi=\varphi_{0} e^{-\beta t} \cos \omega t .\) Find: (a) the angular velocity \(\dot{\varphi}\) and the angular acceleration \(\ddot{\varphi}\) of the body at the moment \(t=0\) (b) the moments of time at which the angular velocity becomes maximum.
4.68. (a) \(\dot{\varphi}(0)=-\beta \varphi_{0}, \quad \ddot{\varphi}(0)=\left(\beta^{2}-\omega^{2}\right) \varphi_{0} ;\) (b) \(\quad t_{n}=\) \(=\frac{1}{\omega}\left(\arctan \frac{\omega^{2}-\beta^{2}}{2 \beta \omega}+n \pi\right),\) where \(n=0,1,2\)