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Question

How can we say that the liquid exerts pressure because of it's weight?

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Solution

Any object with some weight exerts pressure on the surface with which they are in contact with. When we are standing on ground, we are exerting pressure on the ground.
Similarly, fluids have weight and they exert pressure on the base and walls of the container in which they are enclosed. The column of liquid above a surface will exert pressure as per the weight of water column above surface.

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