6 Establish a spatial rectangular coordinate system. 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, we have
$$
\overrightarrow{A M}=\left(0,1, \frac{\sqrt{3}}{2}\right), \overrightarrow{M P}=\left(x, y,-\frac{\sqrt{3}}{2}\right)
$$

Since $A M \perp M P$, we have
$$
\left(0,1, \frac{\sqrt{3}}{2}\right) \cdot\left(x, y,-\frac{\sqrt{3}}{2}\right)=0,
$$

which gives $y=\frac{3}{4}$, which is the equation of the locus of point $P$, and its length within the base disk is $2 \sqrt{1-\left(\frac{3}{4}\right)^{2}}=\frac{\sqrt{7}}{2}$. Therefore, the answer is B.
6 The axial section $SAB$ of a circular cone is an equilateral triangle with a side length of 2, $O$ is the center of the base, $M$ is the midpoint of $SO$, and a moving point $P$ is within the base of the cone (including the circumference). If $AM \perp MP$, then the length of the trajectory formed by point $P$ is ( ).
(A) $\sqrt{7}$
(B) $\frac{\sqrt{7}}{2}$
(C) 3
(D) $\frac{3}{2}$