I am trying to understand what component of the force on an object moving in a regular bowl enables the ball to complete a "semi circle", without any external force being applied. Is the net vertical force a component of inertia?
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There is a net force being applied to the ball (by the bowl). You cannot assume that just because the bowl does not move it implies that the bowl is not exerting a force on the ball. Try it with a paper bowl and see whether your ball can still "complete a semi-circle" or not. Not only is the bowl exerting a force on the ball to prevent it from going straight, whatever is supporting/holding the bowl is exerting exactly that same force plus the force needed to negate the gravitational force on the bowl+ball. If it is a flat table, it does so via friction. You can see this if you use a normal bowl but put it on a slippery surface where it can slide freely.
