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 0.17 m^2 is filled with water. Along, thin tube of cross-sectional area 4.7×10−5 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 m high, the barrel bursts.

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?

To find the pressure of the water on the lid of the barrel, we can use Pascal's law, which states that the pressure exerted by a fluid is transmitted equally in all directions.

The pressure is given by the formula:

Pressure = Force / Area

To find the pressure, we need to find the force exerted by the water on the lid. The force can be calculated by multiplying the pressure by the area.

Given:

Area of the lid (A) = 0.17 m^2

We can assume that the pressure is same at all points on the lid.

Therefore, the pressure exerted by the water on the lid is equal to the pressure exerted by the water column of height 12 m.

The density of water (ρ) is approximately 1000 kg/m^3, and acceleration due to gravity (g) is approximately 9.8 m/s^2.

The pressure is given by the formula:

Pressure = ρ × g × h

Where:
ρ = density of the fluid (water)
g = acceleration due to gravity
h = height of the fluid column

Calculating the pressure:

Pressure = 1000 kg/m^3 × 9.8 m/s^2 × 12 m
Pressure = 117,600 Pa

So, the pressure of the water on the lid of the barrel is 117,600 Pascal.

Now, let's calculate the net force on the lid due to the water pressure.

Net force is given by the formula:

Net Force = Pressure × Area

Given:

Area of the lid (A) = 0.17 m^2

Calculating the net force:

Net Force = 117,600 Pa × 0.17 m^2
Net Force = 19,992 N

So, the net force on the lid due to the water pressure is 19,992 Newtons.

To find the pressure of the water on the lid of the barrel, we can use Pascal's principle, which states that the pressure exerted by a fluid is transmitted uniformly in all directions.

In this scenario, the pressure exerted by the water in the barrel is the same on all sides, so we only need to consider the pressure at the top of the barrel.

We know that pressure is given by the formula:

Pressure = Force / Area

Since the lid of the barrel has an area of 0.17 m^2, we can substitute that value into the formula:

Pressure = Force / 0.17 m^2

To find the force, we can use the fact that the pressure depends on the depth of the fluid. The force exerted by a fluid is given by:

Force = Area * Pressure

Given that the area of the tube is 4.7 × 10^(-5) m^2 and the height of the water column is 12 m, we can calculate the force exerted by the water:

Force = (4.7 × 10^(-5) m^2) * (12 m)

Now, we can substitute the force value into the pressure formula:

Pressure = (Force) / (0.17 m^2)

By evaluating this expression, we can find the pressure of the water on the lid of the barrel.

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

Force = Area * Pressure

In this case, the area is 0.17 m^2, and we can use the pressure value from the previous calculation to determine the net force on the lid.