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A barometer measures changes in air pressure. In the storm glass type of barometer, the nearby air pressure is compared to the pressure of a region of trapped air inside the barometer. If the air pressure at your location increases, more force is applied down on the sighting column liquid and the height of the liquid decreases. When the air pressure drops, the height of the sighting column will rise. Even though temperature can have a large effect on pressure, the barometer is not troubled by this since the air inside and outside the barometer are at nearly the same temperature.

The liquid level in the sighting tube is determined by the pressure difference inside and outside of the barometer. If we think about the position in the sighting tube that is even with the liquid inside the larger vessel, the pressure at that point must be the same as the air pressure inside the vessel. That pressure must also equal the outside air pressure, plus the weight per unit area of the liquid column above that point. So, the air pressure difference is equal to the weight of the supported liquid column divided by the average cross-section area of the liquid column.

Water barometer with internal and external pressures marked.
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