# How did the ball know that a vertical circle follows a slope at the point $A$?

I think the ball has no eyes.

This is my favorite sentence that I have ever seen on Physics SE, and you are absolutely right that the ball has no eyes. The ball does not know that a vertical circle is coming. The only thing the ball knows at any moment is what the track feels like at that point.

For the purpose of calculating the normal force, one needs to know the acceleration of the ball in the direction normal to the surface. That is determined by the curvature of the track at that point. Now, to make this problem simple, the author said that the rest of the track is all a circle with radius $$r$$. So we know at that point A the local radius of curvature is also $$r$$ because point A is on that circle. But it didn't have to be that way. The rest of the track could have been a more complicated shape, and it wouldn't matter. If the local radius of curvature at point A were still $$r$$, the answer would be the same.

To summarize: the author said the rest of the track is a circle to make it easy for you to determine the radius of curvature at the point A. But the ball itself doesn't care about the rest of the track at that moment.