Here is a demonstration pascal used to show the importance of a fluid’s pressure on the fluid’s depth (fig. 9.31): an oak barrel with a lid of area 2 0.20 m is filled with water. a long, thin tube of cross-sectional area 5 2 5.0 10 m   is inserted into a hole at the center of the lid, and water is poured into the tube. when the water reaches 12 m high, the barrel bursts.
(a) what was the weight of the water in the tube?
(b) what was the pressure of the water on the lid of the barrel?
(c) what was the net force on the lid due to the water pressure?

water weight  W = 0.98 N , presurre  P-Po = 1.96 10⁵ Pa  and force  F = 2.47 10⁴ N

Explanation:

Pressure is defined as the relationship between force and area, by means of the equation

               P = F / A

Pascal's principle says that the pressure is passed equally to all directions and points of the fluid. The relationship of pressure with height is

               P = Po + g p h

Where Po if the system is open is the atmospheric pressure. Let's apply these relationships to our case

a) Let's calculate the weight of the water, let's use the expression

               D = m / V

As the density that we take as data for water 1000 Kg / m3, V the volume of the tube, which is a cylinder

We take from the problem the area of the tube and the length of the water column

             L = 20 m

             A1 = 5.0 mm² = 5 10⁻⁶ m²

I think you have several areas, the calculation is the same by changing the value

            V = A1 L

            V = 5 10⁻⁶ 20

            V = 1 10⁻⁴ m3

            W = mg

            D = m / V m = d V

            W = d Vg

            W = 1000 9.8  1 10⁻⁴

            W = 0.98 N

b) Let's calculate the pressure at the bottom of the tube, we take the water density 1000 Kg/m³ as data

            P = Po + d g h

            P = Po + 1000 9.8 20

            P-Po = 1.96 10⁵ Pa

This pressure difference is called gauge pressure

c) The net force

            P = F /A

            F = P A

Barrel area

           R = 0.20 m

           A = π R²

           A = π 0.2²

           A = 0.126 m²

           F = 1.96 10⁵  0.126

           F = 2.47 10⁴ N

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