5. Given $O$ is the circumcenter of $\triangle A B C$, and $D$ is the midpoint of $B C$. If $\overrightarrow{A O} \cdot \overrightarrow{A D}=4, B C=2 \sqrt{6}$, then $A D=$ $\qquad$
5. $\sqrt{2}$.

From the properties of the circumcenter and midpoint, we have
$$
\begin{aligned}
4 & =\overrightarrow{A O} \cdot \overrightarrow{A D}=\frac{1}{2} \overrightarrow{A O} \cdot(\overrightarrow{A B}+\overrightarrow{A C}) \\
& =\frac{1}{4}\left(\overrightarrow{A B}^{2}+\overrightarrow{A C}^{2}\right) .
\end{aligned}
$$

From the properties of the median, we have
$$
\begin{array}{l}
\overrightarrow{A D}^{2}=\frac{1}{4} \times 2\left(\overrightarrow{A B}^{2}+\overrightarrow{A C}^{2}\right)-\overrightarrow{B C}^{2} \\
=\frac{1}{4}(2 \times 16-24)=2 .
\end{array}
$$

Therefore, $A D=\sqrt{2}$.