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Let's take a bottle (no cap) with half its volume filled with water. If we rotate the bottle at some slower average velocity, water does not drop out of it. The same principle works on Roller Coasters. Wikipedia says that the phenomenon of Vertical Loop is used here. What is the physics behind this? (I mean, Which opposes or equilibrates gravity during the vertical loop?)

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Actually, nothing opposes gravity. It's just that the bottle falls as fast as the water (here "fast" refers to acceleration, not velocity) while it's near the top of the loop.

To be specific, when you move anything around in a circle at a constant speed, it has an acceleration of $v^2/R$ directed toward the center of the circle. $v$ is the speed and $R$ is the circle's radius. For something like a bottle moving in a vertical loop, at the top of the loop, that direction is down. Meanwhile, an object subject to gravity naturally accelerates downward with an acceleration of $g = 9.8\ \mathrm{m/s^2}$. So as long as $v^2/R \ge g$, gravity only accelerates the water down as fast as the bottle is already moving.

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