jmc

We can divide the rolling into four phases:

Phase 1: The quarter circle pivots $90^{\circ }$ about point $B$. In this phase, point $B$ does not move.

Phase 2: The quarter circle pivots $90^{\circ }$ about point $C$. In this phase, point $B$ is always $\frac{2}{\pi }$ cm away from point $C$, so its path is a quarter-circle with radius $\frac{2}{\pi }$. The circumference of a circle with radius $\frac{2}{\pi }$ is $2\pi (\frac{2}{\pi })=4$, so $B$ travels $\frac{1}{4}(4)=1$ cm.

Phase 3: The quarter circle rolls along arc $CA$. In this phase, $B$ is always $\frac{2}{\pi }$ away from the ground, so its path is a straight line segment parallel to the ground. From the diagram, we see the length of this line segment is equal to the distance between the original position of $C$ and the new position of $A$. This distance is traced out by arc $CA$ as it rolls. So its length is the length of arc $CA$, which is 1 cm (since it's a quarter of a circle with radius $\frac{2}{\pi }$, a length we've already calculated). So the path of $B$ has length 1 cm.

Phase 4: The quarter circle pivots $90^{\circ }$ about point $A$. As in phase 2, the path of $B$ has length 1 cm.

Putting this together, the path of point $B$ has total length $1+1+1=\boxed{3\text{ cm}}$.