A rod of length $2 \mathrm{~cm}$ makes an angle $\frac{2 \pi}{3} \mathrm{rad}$ with the principal axis of a thin convex lens. The lens has a focal length of $10 \mathrm{~cm}$ and is placed at a distance of $\frac{40}{3} \mathrm{~cm}$ from the object as shown in the figure. The height of the image is $\frac{30 \sqrt{3}}{13} \mathrm{~cm}$ and the angle made by it with respect to the principal axis is $\alpha$ rad. The value of $\alpha$ is $\frac{\pi}{n} r a d$, where $n$ is $\underline{\hspace{2cm}}$ .