PASCAL'S BARREL: HOW IT WORKS AND EXPERIMENTS - PHYSICAL

The barrel Pascal was an experiment conducted by the French scientist Blaise Pascal in 1646 to demonstrate definitively that the liquid pressure propagates identically the same, regardless of the shape of the container.

The experiment consists of filling a barrel with a thin and very tall tube, perfectly adjusted to the filler neck. When the liquid reaches a height of approximately 10 meters (height equivalent to 7 stacked barrels) the barrel bursts due to the pressure exerted by the liquid in the narrow tube.

Illustration of Pascal's Barrel. Source: Wikimedia Commons.

The key to the phenomenon is to understand the concept of pressure. The pressure P that a fluid exerts on a surface is the total force F on that surface divided by the area A of that surface:

P = F / A

How does Pascal's barrel work?

To understand the physical principles of Pascal's experiment, let's calculate the pressure at the bottom of a wine barrel that will fill with water. For greater simplicity of the calculations we will assume it to be cylindrical with the following dimensions: diameter 90 cm and height 130 cm.

As stated, the pressure P at the bottom is the total force F at the bottom, divided by the area A of the bottom:

P = F / A

The area A of the bottom is pi times (π≈3.14) the radius R of the bottom squared:

A = π⋅R ^ 2

In the case of the barrel, it will be 6362 cm ^ 2 equivalent to 0.6362 m ^ 2.

The force F at the bottom of the barrel will be the weight of the water. This weight can be calculated by multiplying the density ρ of water by the volume of water and by the acceleration of gravity g.

F = ρ⋅A⋅h⋅g

In the case of the barrel full of water we have:

F = ρ⋅A⋅h⋅g = 1000 (kg / m ^ 3) ⋅ 0.6362 m ^ 2 ⋅1.30 m⋅10 (m / s ^ 2) = 8271 N.

The force has been calculated in newtons and is equivalent to 827 kg-f, a value quite close to one ton. The pressure at the bottom of the barrel is:

P = F / A = 8271 N / 0.6362 m ^ 2 = 13000 Pa = 13 kPa.

Pressure has been calculated in Pascal (Pa) which is the unit of pressure in the international SI measurement system. One atmosphere of pressure equals 101325 Pa = 101.32 kPa.

Pressure at the bottom of a vertical tube

Let's consider a small tube with an internal diameter of 1 cm and a height equal to that of a barrel, that is, 1.30 meters. The tube is placed vertically with its lower end sealed with a circular cap and is filled with water at its upper end.

Let's first calculate the area of the bottom of the tube:

A = π⋅R ^ 2 = 3.14 * (0.5 cm) ^ 2 = 0.785 cm ^ 2 = 0.0000785 m ^ 2.

The weight of the water contained in the tube is calculated according to the following formula:

F = ρ⋅A⋅h⋅g = 1000 (kg / m ^ 3) ⋅0.0000785 m ^ 2 ⋅1.30 m⋅10 (m / s ^ 2) = 1.0 N.

In other words, the weight of water is 0.1 kg-f, that is to say just 100 grams.

Now let's calculate the pressure:

P = F / A = 1 N / 0.0000785 m ^ 2 = 13000 Pa = 13 kPa.

Amazing! The pressure is the same as that of a barrel. This is the hydrostatic paradox.

Experiments

The pressure at the bottom of Pascal's barrel will be the sum of the pressure produced by the water contained in the barrel itself plus the pressure of the water contained in a narrow tube 9 meters high and 1 cm in diameter that is connected to the mouth. barrel filling.

Figure 2. Blaise Pascal (1623-1662). Source: Palace of Versailles. The pressure at the lower end of the tube will be given by:

P = F / A = ρ⋅A⋅h⋅g / A = ρ⋅g⋅h = 1000 * 10 * 9 Pa = 90000 Pa = 90 kPa.

Notice that in the previous expression the area A is canceled, no matter if it is a large or small area like the tube. In other words, the pressure depends on the height of the surface with respect to the bottom, regardless of the diameter.

Let's add to this pressure the pressure of the barrel itself at the bottom of it:

P _tot = 90 kPa + 13 kPa = 103 kPa.

To find out how much force is being applied to the bottom of the barrel, we multiply the total pressure by the area of the bottom of the barrel.

F _tot = P tot * A = 103000 Pa * 0.6362 m ^ 2 = 65529 N = 6553 kg-f.

In other words, the bottom of the barrel supports 6.5 tons of weight.

Put into practice

Pascal's barrel experiment is easily reproducible at home, provided it is done on a smaller scale. For this, it will not only be necessary to reduce the dimensions, but also to replace the barrel with a glass or container that has less resistance to pressure.

materials

1- A disposable polystyrene cup with a lid. According to the Spanish-speaking country, polystyrene is called in different ways: white cork, styrofoam, polystyrene, foam, anime and other names. These lids are often found at take-out fast food outlets.

2- Plastic hose, preferably transparent 0.5 cm in diameter or smaller and between 1.5 to 1.8 m long.

3- Adhesive tape for packing.

Procedure to carry out the experiment

- Pierce the lid of the polystyrene cup with the help of a drill bit, with a punch, knife or with a cutter, so that a hole is made through which the hose passes tightly.

- Pass the hose through the hole in the lid, so that a small portion of the hose passes into the bowl.

- Neatly seal with tape packing the joint of the hose with the cap on both sides of the cap.

- Place the lid on the jar and seal the joint between the lid and the jar with packing tape, so that no water can escape.

- Put the glass on the floor, and then you have to stretch and raise the hose. It may be helpful to get up using a drop, a stool, or a ladder.

- Fill the glass with water through the hose. It can be helped by a small funnel placed at the tip of the hose to facilitate filling.

When the glass is full and the water level begins to rise through the hose, the pressure increases. There comes a time when the polystyrene glass does not withstand the pressure and bursts, as Pascal demonstrated with his famous barrel.

References

Hydraulic Press. Retrieved from Encyclopædia Britannica: britannica.com.
Hydrostatic Pressure. Recovered from Sensors One: sensorsone.com
Hydrostatic Pressure. Recovered from Oilfield Glossary: glossary.oilfield.slb.com
Pascal's Principle and Hydraulics. National Aeronautics and Space Administration (NASA). Recovered from: grc.nasa.gov.
Serway, R., Jewett, J. (2008). Physics for Science and Engineering. Volume 2. Mexico. Cengage Learning Editors. 367-372.
What is Hydrostatic Pressure: Fluid Pressure and Depth. Recovered from Math and Science Activity Center: edinformatics.com
Well Control School Manual. Chapter 01 Principles of pressure.

PASCAL'S BARREL: HOW IT WORKS AND EXPERIMENTS - PHYSICAL - 2025