Here is a demonstration Pascal used to show the importance of a fluid's pressure on the fluid's depth : An oak barrel with a lid of area 0.25 m^2 is filled with water. Along, thin tube of cross-sectional area 5.5×10−5 m^2 is inserted into a hole at the center of the lid, and water is poured into the tube. When the water reaches 12 {\rm m} high, the barrel bursts.

What was the weight of the water in the tube?
What was the pressure of the water on the lid of the barrel?
What was the net force on the lid due to the water pressure?

Part A

Density of Water is 999.97kg/m3
Volume of tube is (12m*5.5x10-5m2) = 6.6x10-4m3
Density times Volume = mass
999.97kg/m3*6.6x10-4m3 = .65998kg
Weight = m*g = .65998kg*9.8m/s2 = 6.5kg*m/s2 or 6.5N

Part B
P=F/A
6.5kg*m/s2/5.5x10-5m2 = 1.2x10^5kg/m*s2 or 1.2x10^5Pa

Part C
Density times height times gravity times area
phgA
999.97kg/m3*12m*9.8m/s2*.25m2 = 2.9x10^4Kg*m/s2 or 2.9x10^4N

A 5.9 N

B 1.2x10^5 Pa

C 2.4x10^4 N

To find the weight of the water in the tube, we can use the formula: weight = density × volume × gravity.

First, let's find the volume of water in the tube. The volume can be calculated using the formula: volume = cross-sectional area × height.

Given:
Cross-sectional area of the tube (A) = 5.5×10^(-5) m^2
Height of the water in the tube (h) = 12 m

Volume = A × h
Volume = (5.5×10^(-5) m^2) × (12 m)
Volume ≈ 6.6×10^(-4) m^3

Next, we can find the weight of the water using the density of water, which is approximately 1000 kg/m^3, and the acceleration due to gravity, which is approximately 9.8 m/s^2.

Weight = density × volume × gravity
Weight = (1000 kg/m^3) × (6.6×10^(-4) m^3) × (9.8 m/s^2)

Calculating the above expression, we find that the weight of the water in the tube is approximately 6.468 kg.

Now, let's determine the pressure of the water on the lid of the barrel. The formula for pressure is: pressure = force / area.

Given:
Area of the lid (A) = 0.25 m^2

The force can be calculated using the weight of the water calculated earlier.

Force = weight = 6.468 kg × 9.8 m/s^2

Finally, the pressure can be found using the formula:

Pressure = Force / Area

Let's substitute the values into the formula:

Pressure = (6.468 kg × 9.8 m/s^2) / 0.25 m^2

Calculating the above expression, we find that the pressure of the water on the lid of the barrel is approximately 252.7 Pa.

To determine the net force on the lid due to the water pressure, we can use the formula:

Force = pressure × area

Given:
Area of the lid (A) = 0.25 m^2
Pressure (P) = 252.7 Pa

Net force on the lid = Pressure × Area

Substituting the given values into the formula:

Net force = (252.7 Pa) × (0.25 m^2)

Calculating the above expression, we find that the net force on the lid due to the water pressure is approximately 63.18 N.

To find the weight of the water in the tube, we need to use the formula:

Weight = Mass x gravity

First, we need to find the mass of the water in the tube. To do this, we can use the formula:

Mass = Density x Volume

The density of water is given as 1000 kg/m^3, and the volume can be found by multiplying the cross-sectional area of the tube by the height of the water column.

Volume = Cross-sectional area x Height

Substituting the given values, we have:

Volume = (5.5×10−5 m^2) x (12 m)

Now we can substitute the found volume and the known density into the formula for mass:

Mass = (1000 kg/m^3) x [(5.5×10−5 m^2) x (12 m)]

To calculate the weight, we multiply the mass by the acceleration due to gravity, which is 9.8 m/s^2:

Weight = [(1000 kg/m^3) x (5.5×10−5 m^2) x (12 m)] x (9.8 m/s^2)

Now you can plug in the values into a calculator to calculate the weight of the water in the tube.

Next, to find the pressure of the water on the lid of the barrel, we can use the formula:

Pressure = Force / Area

The force in this case is equal to the weight of the water, which we calculated previously. The area is given as 0.25 m^2.

Pressure = [Weight of Water / Volume] / Area

Calculate the pressure using the given values and the weight of the water.

Finally, the net force on the lid due to the water pressure can be found by multiplying the pressure by the area:

Net Force = Pressure x Area

Calculate the net force using the values obtained earlier for pressure and the given area.

By following these steps, you should be able to find the weight of the water in the tube, the pressure of the water on the lid, and the net force on the lid due to the water pressure.