V.2. Hydrostatic pressure.

Hydrostatic pressure is the static pressure inside a liquid in equilibrium due to the weight of the liquid.

Even if the hydrostatic pressure is due to the weight of the liquid, it is exerted in all directions inside it.

We have a liquid at rest in a glass.

h = the height of the liquid column in the vessel

G = the liquid weight = m ∙ g = ρ ∙ V ∙ g = ρ ∙ S ∙ h ∙ g

N' = the force of normal pressure on the bottom of the vessel

N = the normal reaction of the bottom of the vessel

|N'| = |N|= |G|

Hydrostatic pressure formula:

where

ρ = the liquid density

g = gravitational acceleration

h = height of the liquid column above the measured level (liquid depth)

So the hydrostatic pressure depends directly on the density of the liquid and the depth of the liquid. It does not depend on the area of the bottom of the vessel in which the liquid is located.

Hydrostatic pressure is measured with:

When the unevenness of the liquid in the U tube is zero, Δh = 0, the pressure is zero.

The higher the unevenness of the liquid in the U tube, the higher the hydrostatic pressure.

Applications

For example for water (ρ = 1000 kg/m³) the pressure difference is about 10 Pa for each 1 mm (0,001 m) level difference between the 2 branches. We took g ~ 10 N/kg.

Δp = ρ ∙ g ∙ h= 1000 ∙ 10 ∙ 0,001 = 10 Pa.

👀 Experiment: At the same level, the hydrostatic pressure is the same.
🔥 Caution when working with sharp objects! Be careful not to get stung when working with a needle!

Required materials:
PET bottle, compass.

Experiment description:

• Take a 0,5L bottle and fill it with water. Drill it on both sides at the same level.
• Unscrew the plug and notice the two jets of water.

  The two water jets are the same length because, at the same level, the hydrostatic pressure is the same.

👀 Experiment: The hydrostatic pressure increases with depth.
🔥 Caution when working with sharp objects! Be careful not to get stung when working with a needle!

Required materials:
PET bottle, compass.

Experiment description:

• Take a 0,5L bottle, fill it with water and drill it in the same part, one hole above, one in the middle and another towards the bottom of the bottle.
• Unscrew the plug and observe the three jets of water.

  The water jet at the bottom of the bottle is longer than the one above, because the hydrostatic pressure increases with the depth of the liquid.

👀 Experiment: Hydrostatic pressure measurement with manometer.
🔥 Caution when working with sharp objects! Be careful not to get stung when working with a needle!

Required materials:
1,5-2L PET bottle, small hoses, plastic bottle, pen tubes or cylindrical containers, balloon (a piece of a surgical glove), jar elastic, scissors, ruler, compass.

Experiment description:

• Make a U tube by tying two pen bodies with a hose at the bottom. Put colored water in this "U" tube.
• Attach a longer hose to one of the pens.
• Take a plastic bottle and drill the lid so that the hose is forced into it.
• Cut the neck of a 1,5-2 L bottle and put water in it.
• Measures the unevenness of the water in the “U” tube for different levels inside the water.
• Measures the unevenness of the water in the “U” tube at the same level inside the water.

  At the same depth inside the liquid, the unevenness of the colored liquid in the “U” tube is the same, no matter how we orient the capsule membrane, so the hydrostatic pressure is the same. The closer we get to the bottom of the vessel, the greater the unevenness of the liquid in the “U” tube, and the greater the depth of the hydrostatic pressure.

The fundamental principle of hydrostatics:

"The difference in pressure between two points of a liquid in equilibrium is directly proportional to the difference in level at which the two points are."

where

ρ = liquid density (material constant)

g = gravitational acceleration (constant, equal to 9,8 ~ 10 N/kg)

Δh = the level difference inside the liquid at which the two points are located.

Consider a liquid in equilibrium and mentally delimit it with a cylinder (see above drawing). Since the whole liquid is in equilibrium, then the liquid in the delimited cylinder is also in equilibrium. The weight G, the pressure force on the upper surface F₁ and the pressure force on the lower surface F₂ act vertically on it. From the equilibrium condition it results:

G + F₁ = F₂

Because G = ρ ∙ S ∙ g ∙ Δh , F₁ = p₁ ∙ S, F₂ = p₂ ∙ S => Δp = ρ ∙ g ∙ Δh

👀 Experiment: The fundamental principle of hydrostatics.
🔥 Caution when working with sharp objects! Be careful not to get stung when working with a needle!

Required materials: 1,5-2 L PET bottle and 0,5 L PET bottle, a small hose, scissors, compass.

Experiment description:

• Take two bottles, one of 0,5L and 2L and tie them with a hose at the bottom.
• Only fill the small bottle with water.
• What do you notice?

  Water flows from the small bottle to the large bottle.

Experiment conclusion:
Water flows from the small bottle to the large bottle because the water pressure in the small bottle is higher than the pressure in the large bottle, as the water level in the small bottle is higher than the water level in the large bottle. The flow from A to B will take place until the level of the 2 bottles is equal, when the pressure will be the same. The pressure of a liquid in a vessel does not depend on the shape and size of the vessel.

The law of communicating vessels:

"In two or more communicating vessels, the liquid rises to the same level."

Communications Vessel Law Applications

1) Garden sprinkler

2) Teapot

3) The sink siphon holds in its elbow the solid bodies that could clog the sewer.

4) The shuttles allow the movement of ships from a high level of water (upstream) to a lower level (downstream).

5) Level indicator for opaque tanks.

6) Water supply to some homes by placing the water tank at a higher height than the tallest house.

🔓 Solved problems

1. Determine what pressure exerts water with a density of 1000 kg/m³ at a depth of 100 m?

Solution:

We write the problem data:
ρ = 1000 kg/m³
h = 100 m
p = ?

We write the hydrostatic pressure formula:

p = ρ ∙ g ∙ h = 1000 kg/m³ ∙ 10 N/kg ∙ 100 m = 1.000.000 Pa

2. The Dead Sea of Jordan has the lowest water surface in the world (427 m below sea level). It is the saltiest water in the world, 9,6 times saltier than the planet's ocean. The extremely salty content is very unfavorable to life and hence the name Dead Sea. Other records: water with the highest concentration of bromine in the world and the deepest hypersaline lake on earth (306 meters).

Is required: At what depth in the Dead Sea with a density of 1240 kg/m³, the water exerts a pressure of 3.720.000 Pa?

Solution:

We write the problem data:
ρ = 1240 kg/m³
h = ?
p = 3.720.000 Pa

We write the hydrostatic pressure formula and remove the unknown, h:

p = ρ ∙ g ∙ h

3. A cylindrical beaker with a base area S = 20 cm² contains mercury up to a height of 10 cm. Put 400 g of water over the mercury. The density of mercury is 13600 kg/m³, and that of water is 1000 kg/m³.

Calculate: a) The height of the water column, knowing that it is immiscible with mercury (do not mix). b) The pressure exerted by the two liquids on the bottom of the vessel.

Solution:

Write the problem data and turn it into SI:
S = 20 cm² = 0,002 m²
h₁ = 10 cm = 0,1 m
ρ₁ = 13600 kg/m³
ρ₂ = 1000 kg/m³
m₂ = 400 g = 0,4 kg
h₂ = ?
p = ?

With the density formula, we find the volume of water added:

The liquids taking the shape of the vessel, have the same area of the base as of the glass, so the volume of water is:

V₂ = S ∙ h₂

We calculate the hydrostatic pressures of the two liquids:

p₁ = ρ₁ ∙ g ∙ h₁ = 13.600 kg/m³ ∙ 10 N/kg ∙ 0,1 m = 1.3600 Pa

p₂ = ρ₂ ∙ g ∙ h₂ = 1.000 kg/m³ ∙ 10 N/kg ∙ 0,2 m = 2.000 Pa

p = p₁ + p₂ = 13.600 Pa + 2000 Pa = 15.600 Pa

4. In a long-necked bottle, put water up to the middle of the bottle. Then turn the bottle over so that it rests on the stopper. What will the water pressure be like when the bottle is on the bottom compared to the water pressure when the bottle is on the stopper?

Solution:

When the bottle is on the stopper, the height of the water is higher due to the thinner neck than when it is on the bottom. So the water pressure is lower when it is on the bottom than on the stopper. The hydrostatic pressure, for the same liquid, does not depend on the area of the bottom of the vessel, but only on the height of the column of liquid in the vessel.

5. The liquid in a pipette flows only if it is lightly pressed on the plastic tube. Why does the liquid not flow on its own when the pipette is upright?

Solution:

The air pressure above the liquid in the pipette (p) is lower than the atmospheric pressure (p₀) at the tip of the pipette. Atmospheric pressure is equal to the pressure of the air in the pipette plus the pressure of the liquid in the pipette (ρ ∙ g ∙ h).

p₀ = p + ρ ∙ g ∙ h > p