The Pascal barrel was an experiment conducted by the French scientist Blaise Pascal in 1646 to definitively demonstrate that the pressure of the liquid propagates identically to the same, regardless of the shape of the container.
The experiment consists of filling a barrel with a very tall, thin tube, perfectly fitted to the filling mouth. When the liquid reaches a height of approximately 10 meters (height equivalent to 7 piled barrels), the barrel explodes with the pressure exerted by the liquid in the narrow tube.
Pascal barrel illustration. Source: Wikimedia Commons.
The key to the phenomenon is to understand the concept of pressure. The pressure P exerted by a fluid 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 be filled with water. To simplify the calculations, we will assume a cylindrical shape 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 at 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 water density ρ by the water volume and by the gravity acceleration g .
F = ρ⋅A⋅h⋅g
In the case of a 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 was calculated in Newtons and is equivalent to 827 kg-f, a value very 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 was calculated in Pascal (Pa), which is the unit of pressure in the international SI measurement system. One atmosphere of pressure is equal to 101325 Pa = 101.32 kPa.
Pressure at the bottom of a vertical tube
Consider a 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 the lower end sealed with a circular cap and filled with water at the 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.
That is, the weight of water is 0.1 kg-f, that is, only 100 grams.
Let’s calculate the pressure now:
P = F / A = 1 N / 0.0000785 m^2 = 13000 Pa = 13 kPa.
Incredible! The pressure is the same as that of a barrel. This is the hydrostatic paradox.
Experiences
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 of filling the barrel.
P = F / A = ρ⋅A⋅h⋅g / A = ρ⋅g⋅h = 1000 * 10 * 9 Pa = 90000 Pa = 90 kPa .
Note that in the previous expression area A has been canceled, regardless of whether it is a large or small area like the tube. In other words, pressure depends on the height of the surface in relation to the depth, regardless of diameter.
Let’s add to this pressure the pressure of the barrel itself at the bottom:
P tot = 90 kPa + 13 kPa = 103 kPa .
To find out how much force is being applied to the bottom of the keg, we multiply the total pressure by the area of the bottom of the keg.
F tot = P tot * A = 103000 Pa * 0.6362 m^2 = 65529 N = 6553 kg-f .
That is, the bottom of the barrel supports 6.5 tons of weight.
Implementation
The Pascal barrel experiment is easily reproducible at home, as long as it is done on a smaller scale. For this, it will be necessary not only to reduce the dimensions, but also to replace the barrel with a cup 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 several ways: white cork, unicel, polyespan, foam, anime and other names. These lidded glasses are usually available from fast food outlets.
2- Plastic hose, preferably transparent, 0.5 cm in diameter or smaller and between 1.5 and 1.8 m in length.
3- Adhesive tape for packaging.
Procedure for performing the experiment
– Drill the cover of the polystyrene glass with the help of a drill, a hole punch, knife or a cutter, so that a hole is made through which the hose passes firmly.
– Pass the hose through the hole in the cap so that a small part of the hose passes into the container.
– Seal carefully with adhesive tape and secure the hose joint with the cap on both sides of the cap.
– Place the lid on the glass and also seal the connection between the lid and the glass with adhesive tape, so that water cannot escape.
– Place the cup on the floor and then you will have to stretch and lift the hose. It can be helpful to climb using an incline, a sidewalk, or a ladder.
– Fill the glass with water through the hose. It can be helped with a small funnel placed at the end of the hose to facilitate filling.
When the glass is full and the water level starts to rise through the hose, the pressure increases. There comes a time when polystyrene glass cannot withstand the pressure and explosions, as Pascal showed with his famous barrel.
