A square rotates around a moving pivot

Just continuously rotate and translate the entire system such that the square isn't moving. Now the point's direction is rotating at the original rate of the square--a constant rate--and the point is moving at a constant speed. It is tracing out a circle on the square. As we speed up its rate of rotation, the point's path radius decreases. Thus, we want to find the rate of rotation to yield the largest path radius. The point started in the center of the square, and began moving perpendicular to one face of the square; the path of the point is tangent to $l$ which passes through the center of the square and the path passes through the center itself. Our desired radius is $\frac14$ because the circle with these characteristics is only tangent to one of the faces of the squares; any larger and circle goes off the square. The point travels at 1m/s and thus does so around the circle at 1m/s. After one round around the circumference, the angle of its direction will have changed by $2\pi$ rad. These rates of changes are constant and so we can find them with only two data points. The variables are self explanatory: $\frac{d\theta}{dt}=\frac{2\pi (radians)}{\frac{2\pi r(meters)}{1(meter/second)}}=\frac{1}{r}(radians/second)=4(radians/second)$ .

TL;DR Wheel rolling on ground: point of contact is point in question and its path is the wheel