4. As shown in the figure, the vertex $B$ of $\triangle A B C$ is at the center of the unit circle, and $A, C$ are on the circumference, $\angle A B C=2 \alpha(0<\alpha<\frac{\pi}{3})$. Now, $\triangle A B C$ is rotated counterclockwise within the circle, with the specific method as follows: First, rotate around $A$ to make $B$ fall on the circumference; second, rotate around $B$ to make $C$ fall on the circumference; third, rotate around $C$ to make $A$ fall on the circumference, and so on until the 100th rotation. Then, the total length of the path traveled by point $A$ is
(A) $22 \pi(1+\sin \alpha)-66 \alpha$;
(B) $22 \pi+\frac{68}{3} \pi \sin \alpha-66 \alpha$;
(C) $\frac{67}{3} \pi$;
(D) $33 \pi-66 \alpha$.
Answer ( )
4. When $\triangle A B C$ rotates counterclockwise in the circle, sequentially centered at $A, B, C$ (i.e., at $A, B_{1}, C_{2}$ in the diagram), until point $B$ returns to the center of the circle, point $A$ traces out the circular arcs $\overparen{A A_{1}}$ and $\overparen{A_{1} A_{2}}$, with radii 1 and the side length $A C = 2 \sin \alpha$, respectively.

Since $\triangle A B_{1} B$ and $\triangle B_{1} C_{2} B$ are equilateral triangles, and $\angle A_{1} B_{1} C_{2} = 2 \alpha$, the central angle corresponding to $\overparen{A A_{1}}$ is $\left(\frac{2}{3} \pi - 2 \alpha\right)$. When $\triangle A_{1} B_{1} C_{2}$ rotates for the third time with $C_{2}$ as the center, the side $C_{2} B_{1}$ rotates by $\frac{\pi}{3}$ to reach $C_{2} B$, so the central angle corresponding to $\overparen{A}_{1} A_{2}$ is $\frac{\pi}{3}$. Thus, each time $\triangle A B C$ rotates three times, the distance traveled by point $A$ is:
$$
\left(\frac{2}{3} \pi - 2 \alpha\right) \cdot 1 + \frac{\pi}{3} \cdot 2 \sin \alpha = \frac{2}{3} \pi (1 + \sin \alpha) - 2 \alpha.
$$

Therefore, when $\triangle A B C$ rotates 100 times, the distance traveled by point $A$ is:
$$
\begin{aligned}
s & = \frac{100-1}{3} \left[ \frac{2}{3} \pi (1 + \sin \alpha) - 2 \alpha \right] \\
& = 22 \pi (1 + \sin \alpha) - 66 \alpha.
\end{aligned}
$$