5. Given $O$ is the circumcenter of $\triangle A B C$, $D$ is the midpoint of $B C$, if $\overrightarrow{A O} \cdot \overrightarrow{A D}=4, B C=2 \sqrt{6}$, then $A D=$
$$
\overrightarrow{A O} \cdot \overrightarrow{A D}=\frac{1}{2}(\overrightarrow{A O} \cdot \overrightarrow{A B}+\overrightarrow{A O} \cdot \overrightarrow{A C})=\frac{1}{4}\left(c^{2}+b^{2}\right)=4 \Rightarrow b^{2}+c^{2}=16
$$

From the median length formula of a triangle, we get $A D^{2}+\frac{1}{4} a^{2}=\frac{1}{2}\left(b^{2}+c^{2}\right) \Rightarrow A D=\sqrt{2}$.