5. Given point $A$ is a point inside the unit circle with center $O$, satisfying $|\overrightarrow{O A}|=\frac{1}{2}, B, C$ are any two points on the unit circle $O$, then the range of $\overrightarrow{A C} \cdot \overrightarrow{B C}$ is $\qquad$ .
5. $\left[-\frac{1}{8}, 3\right]$.
(Method 1) First fix $\overrightarrow{C B}$. Let the projections of $A, O$ on $C B$ be $H, M$, respectively, and the projection of the circle with center $O$ and radius $\frac{1}{2}$ on $C B$ be the line segment $P Q$. Then
$$
\overrightarrow{C A} \cdot \overrightarrow{C B}=\overrightarrow{C H} \cdot \overrightarrow{C B}=(\overrightarrow{C M}+\overrightarrow{M H}) \cdot \overrightarrow{C B}=\frac{1}{2} \overrightarrow{C B}^{2}+\overrightarrow{M H} \cdot \overrightarrow{C B}
$$

Considering that $H$ is on the line segment $P Q$, the maximum value of $C A \cdot C B$ is $\frac{1}{2}|C B|^{2}+\frac{1}{2}|C B|$, and the minimum value is $\frac{1}{2}|C B|^{2}-\frac{1}{2}|C B|$.
Since $0 \leqslant|C B| \leqslant 2$, the maximum value of $\overrightarrow{C A} \cdot \overrightarrow{C B}$ is 3, and the minimum value is $-\frac{1}{8}$.
(Method 2) Unify the starting point as $O$, then
$$
\begin{aligned}
\overrightarrow{A C} \cdot \overrightarrow{B C} & =(\overrightarrow{O C}-\overrightarrow{O A}) \cdot(\overrightarrow{O C}-\overrightarrow{O B}) \\
& =\overrightarrow{O C}^{2}-\overrightarrow{O A} \cdot \overrightarrow{O C}+\overrightarrow{O A} \cdot \overrightarrow{O B}-\overrightarrow{O C} \cdot \overrightarrow{O B} \\
& =\overrightarrow{O C}^{2}+\frac{1}{2}\left((\overrightarrow{O A}+\overrightarrow{O B}-\overrightarrow{O C})^{2}-\overrightarrow{O A}^{2}-\overrightarrow{O B}^{2}-\overrightarrow{O C}^{2}\right) \\
& =\frac{1}{2}(\overrightarrow{O A}+\overrightarrow{O B}-\overrightarrow{O C})^{2}-\frac{1}{8} .
\end{aligned}
$$

Considering that $\overrightarrow{O B}-\overrightarrow{O C}$ can represent any vector with length from 0 to 2, the length of $\overrightarrow{O A}+\overrightarrow{O B}-\overrightarrow{O C}$ ranges from $\left[0, \frac{5}{2}\right]$, so the required range is $\left[-\frac{1}{8}, 3\right]$.