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So I just noticed that when I filled my bucket with water until it overflowed and then I turned the faucet off, the water kept dripping for like 20 seconds. Why does this happens? Shouldn't it have stopped dripping sooner after I turned the faucet off?

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If you are talking about the water overflowing from the bucket, it is because water has a surface tension. This lets the surface, or skin of the water rise slightly above the rim of the bucket before the gravity pulls the water enough to break the surface tension and start it flowing over the edge. Once water stops flowing into the bucket it will still flow out for a short while until it's surface tension equals the gravity pulling it over the edge.

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There are two possible scenarios here, which I don't know which it is unless I have more details.

One is where you fill slowly until overflow just begins to happen; this results in Adrian's answer, which I will elaborate on a bit:

The surface tension of water means that the water level can rise above the level of the bucket edge by forming a "bubble" above the lip; Laplace pressure, pressure due to the curvature of such a surface (surface geometry looks like the illustration here), counteracts gravitational pressure, preventing the water from spilling. Once you overfill enough, this pressure is no longer sufficient, the surface breaks, and water begins to spill; once it does, the vertical curvature at the overflow point reduces (to form the outflow stream), weakening the Laplace pressure, which results in the water continuing to spill out until the level lowers enough that the gravitational pressure is lower than the new, lowered Laplace pressure.

The other scenario is where you fill the water fast. Here, it's just due to the inertial and viscosity forces of the fluid; there must be a surface level imbalance for fluid with weight/viscosity to flow (think a river, flowing from high to low), which means there is a gradient in surface level from where you're filling the bucket to where it is spilling out (flow requires a pressure gradient, which results in a surface level gradient); once you stop the inflow, the fluid will leak until the surface is level, which takes some time. At low flows, the surface level gradient is rougly proportional to flow, which means that technically, without any other effects, the flow will only exponentially decay rather than actually stop; however, surface tension as mentioned above will eventually cause water to abruptly stop flowing.

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