13.299. Two points rotate uniformly along two concentric circles. One of them completes a full revolution 5 seconds faster than the other, and therefore manages to make two more revolutions per minute. Let the rays directed from the center of the circle to these points coincide at the beginning of the motion. Calculate the magnitude of the angle between the rays after $1 \mathrm{s}$.
Solution.

Let $V_{1}, V_{2}$ be the rotational speeds of the points, $l$ be the length of the circumference, and $x$ be the length of the arc between the points after 1 second of rotation. Then the angle between the rays will be $\alpha=\frac{x \cdot 360^{\circ}}{l}$ (1). According to the problem, we have the system $\left\{\begin{array}{l}\frac{l}{V_{2}}-\frac{l}{V_{1}}=5, \\ 60\left(V_{1}-V_{2}\right)=2 l,\end{array}\right.$ from which $V_{1}=\frac{l}{10}, V_{2}=\frac{l}{15}$.

If the points rotate in the same direction, then $x=V_{1}-V_{2}=\frac{l}{30}$ and $\alpha=12^{\circ}$; if in opposite directions, then $x=V_{1}+V_{2}=\frac{l}{6}$ and $\alpha=60^{\circ}$.

Answer: 12 or $60^{\circ}$.