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Suppose there is no air trapped in the syringe. When you put small amount of weight on the pan, the piston stays in its position. If you pull the piston down and release, it will be pushed up because the force by the atmosphere is more than the weight you put. But if the weight is equal to the force by the atmosphere, and you pull the piston little bit and release, the piston will stay wherever it is released. By measuring the inner area of cross section of the syringe, you can get the atmospheric pressure by the formula Atmospheric Pressure = Weight/Area.
You need a syringe with one end closed and very little air trapped in it, an identical extra syringe, support system for keeping the syringe fixed in vertical position, a pan suspended from the lower end of the syringe barrel, known weights.
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