Jesse cuts a circular paper disk of radius $12$ along two radii to form two sectors, the smaller having a central angle of $120$ degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger one? $\textbf{(A)}\ \frac{1}{8}\qquad$ $\textbf{(B)}\ \frac{1}{4}\qquad$ $\textbf{(C)}\ \frac{\sqrt{10}}{10}\qquad$ $\textbf{(D)}\ \frac{\sqrt{5}}{6}\qquad$ $\textbf{(E)}\ \frac{\sqrt{5}}{5}$