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Mechanical Oscillations

Problem 4.68

A body performs torsional oscillations according to the law φ=φ0e−βtcos⁡ωt.\varphi=\varphi_{0} e^{-\beta t} \cos \omega t . Find: (a) the angular velocity φ˙\dot{\varphi} and the angular acceleration φ¨\ddot{\varphi} of the body at the moment t=0t=0 (b) the moments of time at which the angular velocity becomes maximum.

Reveal Answer
4.68. (a) φ˙(0)=−βφ0,φ¨(0)=(β2−ω2)φ0;\dot{\varphi}(0)=-\beta \varphi_{0}, \quad \ddot{\varphi}(0)=\left(\beta^{2}-\omega^{2}\right) \varphi_{0} ; (b) tn=\quad t_{n}= =1ω(arctan⁡ω2−β22βω+nπ),=\frac{1}{\omega}\left(\arctan \frac{\omega^{2}-\beta^{2}}{2 \beta \omega}+n \pi\right), where n=0,1,2n=0,1,2
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