We have two dots $d_1,d_2$ moving on circles $C_1, C_2$ with radii $r_1, r_2$. The circles are moving at speed of $s_1, s_2$. A line is drawn between $d_1$ and $d_2$. When does this line have some segment inside the smaller of the two circles?
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Assuming that the suffix 1 refers to the inner circle, and $s_1$ is the angular speed of the inner circle, then relative to the point on the outer circle, the point on the inner circle completes one revolution in time $\tau=\frac{2\pi}{\omega}$, where $\omega=|s_1-s_2|$. The fraction of this time when the joining chord does not intersect the inner circle is given by $T=\frac{\theta}{\pi}\tau$, where $\theta=\arccos(\frac{r_1}{r_2})$, from which you can obtain an expression for the time when the chord intersects the inner circle, i.e. $\tau-T$