Doubts regarding block sliding on hemisphere

Lets change the problem a little. Suppose the body was tied to the hemisphere like a rollercoaster is tied to a track. An suppose we slide the body around the hemisphere at a slow constant speed. That is, we also have a force tangent to the hemisphere holding the body back.

When the body starts out at the top, it is sitting on the hemisphere. It doesn't fall through because the hemisphere is made of some rigid material that exerts an upward force just strong enough to keep the body from falling through. Since the body moves in a circle at constant speed, the total force is centripetal. $F = mv^2/r$.

As the body slides below the equator, the body doesn't fall off because the track holds it. The total force is still centripetal. But now the normal force is toward the center. If it wasn't for the inward normal force, the body would fall off.

If you start from the top at a faster speed, the body will stay on the hemisphere if gravity larger than the centripetal force needed to move along the hemisphere. The outward normal force will oppose gravity and reduce the total force to exactly what is needed to follow the hemisphere.

As the body slides down, the component of gravity toward the center gets smaller. There will be a point where that component isn't bigger than the centripetal force needed to follow the hemisphere. At that point, the body will follow a straighter curve than the hemisphere, and fly off.