7. The axial section $S A B$ of the circular cone is an equilateral triangle with a side length of 2, $O$ is the center of the base, $M$ is the midpoint of $S O$, and the moving point $P$ is within the base of the cone (including the circumference). If $A M \perp M P$, then the length of the trajectory formed by point $P$ is $\qquad$
7. $\frac{\sqrt{7}}{2}$.
Establish a rectangular coordinate system in space.
Let $A(0,-1,0), B(0,1,0)$,
$S(0,0, \sqrt{3}), M\left(0,0, \frac{\sqrt{3}}{2}\right), P(x, y, 0)$.
Thus, $\overrightarrow{A M}=\left(0,1, \frac{\sqrt{3}}{2}\right), \overrightarrow{M P}=\left(x, y,-\frac{\sqrt{3}}{2}\right)$.
Since $A M \perp M P$, we have
$$
\left(0,1, \frac{\sqrt{3}}{2}\right) \cdot\left(x, y,-\frac{\sqrt{3}}{2}\right)=0 \Rightarrow y=\frac{3}{4},
$$

This is the equation of the locus of point $P$, and its length within the circular disk at the base is $2 \sqrt{1-\left(\frac{3}{4}\right)^{2}}=\frac{\sqrt{7}}{2}$.