11. (2004 National College Entrance Examination) Given a quadrilateral pyramid $P-ABCD$, $PB \perp AD$, the lateral face $PAD$ is an equilateral triangle with side length equal to 2, the base $ABCD$ is a rhombus, and the dihedral angle between the lateral face $PAD$ and the base $ABCD$ is $120^{\circ}$.
(I) Find the distance from point $P$ to the plane $ABCD$;
(II) Find the size of the dihedral angle between the faces $APB$ and $CPB$.
11. ( I ) As shown in the figure, construct $P O \perp$ plane $A B C D$, with the foot of the perpendicular being point $O$. Connect $O B, O A, O D, O B$ intersects $A D$ at point $E$, and connect $P E$. Since $A D \perp P B$, it follows that $A D \perp O B$. Also, since $P A = P D$, then $O A = O D$.
Thus, $O B$ bisects $A D$, and point $E$ is the midpoint of $A D$, so $P E \perp A D$. Therefore, $\angle P E B$ is the plane angle of the dihedral angle formed by plane $P A D$ and plane $A B C D$.
Thus, $\angle P E B = 120^{\circ}$, and hence $\angle P E O = 60^{\circ}$.
From the given information, we can find that $P E = \sqrt{3}$.
Therefore, $P O = P E \cdot \sin 60^{\circ} = \sqrt{3} \times \frac{\sqrt{3}}{2} = \frac{3}{2}$,
which means the distance from point $P$ to plane $A B C D$ is $\frac{3}{2}$.
( II ) As shown in the figure, establish a rectangular coordinate system, with $O$ as the origin and the $x$-axis parallel to $D A$. $P\left(0,0, \frac{3}{2}\right), B\left(0, \frac{3 \sqrt{3}}{2}, 0\right)$, and the coordinates of the midpoint $G$ of $P B$ are $\left(0, \frac{3 \sqrt{3}}{4}, \frac{3}{4}\right)$. Connect $A G$. Also, $A\left(1, \frac{\sqrt{3}}{2}, 0\right), C\left(-2, \frac{3 \sqrt{3}}{2}, 0\right)$, from which we get:
$$
\overrightarrow{G A}=\left(1,-\frac{\sqrt{3}}{4},-\frac{3}{4}\right), \overrightarrow{P B}=\left(0, \frac{3 \sqrt{3}}{2},-\frac{3}{2}\right), \overrightarrow{B C}=(-2,0,0) .
$$

Thus, $\overrightarrow{G A} \cdot \overrightarrow{P B}=0, \overrightarrow{B C} \cdot \overrightarrow{P B}=0$.
Therefore, $\overrightarrow{G A} \perp \overrightarrow{P B}, \overrightarrow{B C} \perp \overrightarrow{P B}$, and the angle $\theta$ between $\overrightarrow{G A}$ and $\overrightarrow{B C}$ is the plane angle of the dihedral angle we are looking for.
Thus, $\cos \theta=\frac{\overrightarrow{G A} \cdot \overrightarrow{B C}}{|\overrightarrow{G A}||\overrightarrow{B C}|}=-\frac{2 \sqrt{7}}{7}$,
so the size of the dihedral angle we are looking for is $\pi-\arccos \frac{2 \sqrt{7}}{7}$.