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Name: Chris H.
Status: student
Age: 17
Location: N/A
Country: N/A
Date: Thursday, November 28, 2002


Question:
I have a bottle with two holes (one in the top one in the bottom), I am performing an experiment showing how when the hole in the top is covered (air hole) the water will not flow out the small hole in the bottom. As there becomes less and less water in the bottle the time delay between covering the hole at the top of the bottle and water coming out the bottom increases - why?


Replies:
The underlying reason is in the way pressure and volume are related in a gas.

Let us start with the top hole uncovered, with air flowing into the bottle and water flowing out the bottom hole. Now cover the top hole and let us look at the forces on the cylinder of water that extends from the bottom hole up to the top surface of the water inside the bottle.

1) Atmospheric pressure below the bottom hole, acting over the area of the bottom hole; this is your only upward force and it is equal to P_o * A, where P_o is the pressure outside the bottle, and A is the area of the hole.

2) The gravitational force W of the cylinder of water; this is a downward force, obviously.

3) Air trapped in the bottle, with a pressure that will vary as water leaves the bottle (with the top plugged). This pressure also acts over the area of the hole, and it produces a downward force equal to P_i * A, where P_i is the pressure in the bottle. (Do not get sucked in by the notion of some sort of vacuum inside the bottle sucking the water in. The air inside the bottle pushes water down. Period.) All sideways forces on the cylinder cancel out, and are therefore ignorable. Initially, the air in the bottle is at atmospheric pressure, and its force cancels the upward force from the air under the bottle. The remaining force is the weight of the water, and this causes water to move down through the hole. As water leaves the bottle, the air inside increases in volume, and this causes its pressure to decrease. If the amount of air stays constant, and its temperature does not change, the equation that relates pressure and volume is

Pressure * Volume = constant,

so, as the volume increases the pressure decreases. As the pressure decreases, the total downward force on the cylinder decreases. Here is an expression for that force:

Force = P_o * A + W - P_i * A.

If P_i decreases to the point that the total downward force is zero, then water will stop flowing out the bottom hole.

How long will that take? Let us assume for simplicity that we can ignore the changing height of the water cylinder. Let us assume for concreteness that the water stops flowing when the pressure inside the bottle is just half of the pressure outside. If we start with a small volume of air, it does not take much to double that volume (halve the pressure) -- only a small quantity of water has to leave. But if we have a large air volume in the bottle, then a lot of water must exit in order for that air volume to double. This is why it takes longer for the water to stop flowing as the amount of water in the bottle decreases.

Tim Mooney


Chris,

I have not performed this experiment myself but I am not surprised that there is a relationship between the amount of water in the jar and how long it takes for the flow to stop. Actually, it is probably related to the amount of air in the bottle. In order for the flow to stop, the atmospheric pressure outside the bottle must equal the force due to the weight of the water in the bottle plus the force due to the air inside the bottle. When you cover the bottle the water continues to pour out until the air pressure inside the bottle drops sufficiently that the external air pressure can hold the water in. If there is very little air space it does not take long for enough water to pour out, increasing the air volume in the bottle, to achieve this balance. When there is lots of air to begin with much more water must flow out before the air pressure inside drops enough to hold the water in.

Greg Bradburn



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