Question Number 190257 by jlewis last updated on 30/Mar/23
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Commented by jlewis last updated on 30/Mar/23
Commented by mahdipoor last updated on 31/Mar/23
$${x}={Asin}\left({Bt}+\phi\right)\:\:\:\Rightarrow \\ $$$$\overset{..} {{x}}+{wx}=−{AB}^{\mathrm{2}} {sin}\left({Bt}+\phi\right)+{w}^{\mathrm{2}} {Asin}\left({Bt}+\phi\right)=\mathrm{0} \\ $$$$\Rightarrow{B}={w}\:\Rightarrow \\ $$$${x}={Asin}\left({wt}+\phi\right)\:\Rightarrow\begin{cases}{{x}\left(\mathrm{0}\right)=\mathrm{0}\:\Rightarrow\:\phi=\mathrm{0}\:{or}\:\pi}\\{\overset{.} {{x}}\left(\mathrm{0}\right)={v}\:\Rightarrow{v}={Awcos}\phi}\end{cases} \\ $$$${if}:\:\:\:\:\:{x}={Asin}\left({wt}+\phi\right) \\ $$$${v}>\mathrm{0}\:\Rightarrow\:\phi=\mathrm{0}\:\:\:\:{A}=\frac{{v}}{{w}} \\ $$$${v}<\mathrm{0}\:\Rightarrow\:\phi=\pi\:\:\:\:{A}=\frac{−{v}}{{w}} \\ $$$${v}=\mathrm{0}\:\Rightarrow\:{A}=\mathrm{0} \\ $$