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# In the simple AC circuit shown in Figure , $R=70.0 \Omega$ and $\Delta v=\Delta V_{\max } \sin \omega t$. (a) If $\Delta v_R=0.250 \Delta V_{\max }$ for the first time at $t=0.0100 \mathrm{~s}$, what is the angular frequency of the source? (b) What is the next value of $t$ for which $\Delta v_R=0.250 \Delta V_{\max } ?$

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$\textbf{\large Part A}$

The angular frequency is determined from the relation for the voltage of the AC circuit:

\begin{align*} &V=V_{0}\sin \omega t\\ &0.25V_{0}=V_{0}\sin 0.01\omega\\ \omega&=\dfrac{\arcsin(0.25)}{0.01}\:\text{s}^{-1}\\ &=\boxed{25.3\:\text{s}^{-1}} \end{align*}

$\textbf{\large Part B}$

To determine when is the next time the given voltage occurs we use the properties of the sine function:

\begin{align*} &\sin \omega t=\sin (\pi-\omega t)\\ &\sin (0.01\cdot25.3)=\sin (\pi-25.3t)\\ t&=\dfrac{\pi-0.253}{25.3}\:\text{s}\\ &=\boxed{0.114\:\text{s}} \end{align*}

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