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In capillary action the water for example rises in a small-diameter-glass tube due to that the cohesive and adhesive forces are larger than gravity and fall if they were less than gravity .The question that comes to mind is that why are we neglecting the atmospheric pressure or force in this action of rising or falling , although , it is really significant as same as the weight force effect?

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We neglect it because that pressure acts simultaneously on all parts of the system, which means that particular set of forces cancels out.

That's a simplification inasmuch as it is true that if the two ends of the capillary tube are at different heights, they will experience different amounts of ambient pressure. However, that hydrostatic difference is of order ~several inches of air (not water or mercury!) which, for practical purposes, is unmeasurable. So, we neglect it and still get useful and accurate results from measurements on capillary tubes, j-tube manometers, etc.

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