Chapter 5: Properties of Hydrostatics

• Pressure at a point and its unit; why a pointed heel can press more than an elephant's foot.
• Pressure at a point inside a liquid: P = P_a + ρgh, with its consequences (connecting vessels, dams, divers).
• Balance of immiscible liquids in a U-tube; finding the relative density of oil.
• Atmospheric pressure, the mercury barometer (Torricelli), and units of pressure.
• The manometer.
• Applications of pressure (blood pressure, tyre pressure).
• Pascal's principle and the hydraulic press, mechanical advantage, hydraulic brakes and lifts.

After studying fluids in motion (hydrodynamics), we study fluids at rest (hydrostatics): how a liquid exerts pressure at every point inside it, how we measure the pressure of the atmosphere and of gases, and how pressure applied to a liquid is transmitted.

Pressure at a point is the average force acting perpendicularly on a unit area surrounding that point. If a force F acts perpendicularly on a surface of area A, the pressure is

P = F / A

• Unit: newton per square metre, N/m², called the pascal (Pa): 1 Pa = 1 N/m².
• Dimensional formula: ML⁻¹T⁻².
• Pressure is a scalar quantity (the force it produces on a surface is always perpendicular to that surface).

Pressure increases when the force increases or when the area decreases.

• Which presses harder on the ground, an elephant's foot or a woman's pointed heel?
• Because the area of a pointed heel is tiny, the pressure under it can be much greater than under an elephant's broad foot, even though the elephant's weight is far larger.

Example 1. A 60 kg person stands on both feet, each of contact area 200 cm². (g = 9.8 m/s²) Weight = 60 × 9.8 = 588 N; total area = 400 cm² = 0.04 m². P = 588/0.04 = 14 700 N/m². Standing on one foot doubles the pressure to 29 400 N/m².

Example 2. A heel of area 1 cm² carries 300 N: P = 300/10⁻⁴ = 3 × 10⁶ N/m². A foot of area 0.04 m² carrying 5000 N gives only 1.25 × 10⁵ N/m².

Example 3 (a block). A solid block 4 cm × 10 cm × 25 cm has density 2500 kg/m³ (take g = 10 m/s²). Its volume is 1000 cm³ = 10⁻³ m³; mass = 2.5 kg; weight = 25 N.

• Largest pressure (smallest face 4 × 10 cm² = 4 × 10⁻³ m²): P = 25 / 4 × 10⁻³ = 6250 N/m².
• Smallest pressure (largest face 10 × 25 = 250 cm² = 0.025 m²): P = 25/0.025 = 1000 N/m².

Why do we use a sharp knife, and wide tracks on a tractor? Sharp edge → small area → large pressure (cuts easily). Wide tracks → large area → small pressure (does not sink in soft ground).

• Push a cork under water and release it: it rises back to the surface.
• So the water exerts an upward force on the submerged cork; this force comes from the difference in pressure acting on the bottom and the top of the cork.

• At any point inside a liquid the pressure acts in all directions.
• The force produced by liquid pressure on a surface is always perpendicular to that surface.
• The pressure increases with depth.

• Imagine a horizontal plate of area A at depth h below the free surface of a liquid of density ρ.
• The plate carries a vertical column of liquid above it, of height h and cross-section A.

• Volume of the column = A h
• Mass of the column = ρ A h
• Weight of the column = F_g = ρ A h g

• Because the liquid is at rest and incompressible, the force on the plate due to the liquid equals the weight of this column.
• So the pressure due to the liquid alone is

P = F/A = ρ A h g / A → P = ρ g h (N/m²)

If the liquid surface is open to the atmosphere (atmospheric pressure P_a), this pressure is added, so the total (absolute) pressure at depth h is

P = P_a + ρ g h

The quantity ρgh is the gauge pressure — the pressure due to the liquid only.

The pressure at a point depends on:

• the depth h (greater depth → greater pressure),
• the density of the liquid ρ (greater density → greater pressure at the same depth),
• g and the pressure at the surface P_a. It does not depend on the shape or width of the container nor on the area of its base, nor on the amount of liquid.

1. All points at the same horizontal level inside the same liquid at rest have the same pressure.
2. Connecting vessels: a liquid poured into vessels connected at the bottom rises to the same level in all of them, whatever their shapes (the pressure at their common base level must be the same). So the sea level is the same for all connected seas.
3. Dams are built thicker at the base to withstand the greater pressure at greater depths.
4. Divers wear protective suits. Near the surface, a diver blows air into the sinuses to equalise the pressure. At great depths (hundreds of metres) a pressurised diving suit and helmet protect him from the huge external pressure.

Example 1 (lake). Find the gauge pressure and the total pressure at a depth of 12 m in fresh water. (ρ = 1000 kg/m³, g = 9.8 m/s², P_a = 1.013 × 10⁵ N/m²) ρgh = 1000 × 9.8 × 12 = 1.176 × 10⁵ N/m²; total P = 1.013 × 10⁵ + 1.176 × 10⁵ = 2.189 × 10⁵ N/m² (about 2.2 atm).

Example 2 (force on a tank bottom). An oil tank (ρ = 850 kg/m³) is filled to a depth of 2.5 m. Its base area is 4 m². (g = 10 m/s², P_a = 1.013 × 10⁵ N/m²) ρgh = 850 × 10 × 2.5 = 21 250 N/m². Total pressure P = 1.013 × 10⁵ + 0.2125 × 10⁵ = 1.2255 × 10⁵ N/m². Total force on the base = P × A = 1.2255 × 10⁵ × 4 = 4.9 × 10⁵ N. (Force due to the oil alone: 21 250 × 4 = 85 000 N.)

Example 3 (diver). A diver is at 30 m in sea water (ρ = 1030 kg/m³). ρgh = 1030 × 9.8 × 30 = 3.03 × 10⁵ N/m²; total = 4.04 × 10⁵ N/m² ≈ 4 atm.

Example 4 (depth from pressure). At what depth in water is the total pressure 3 atm if the surface pressure is 1 atm? The extra 2 atm = 2 × 1.013 × 10⁵ = ρgh → h = 2.026 × 10⁵/(1000 × 9.8) = 20.7 m.

• Pour some water into a U-shaped tube.
• Then pour oil (which does not mix with water) into one arm until the oil/water separating surface is at A in that arm.
• Let the oil column above the separating level have height h_o and the water column above the same horizontal level in the other arm have height h_w.
• The points at the same level in the same liquid have the same pressure.
• Applying this to the two points at the level of the separating surface:

P_a + ρ_o g h_o = P_a + ρ_w g h_w → ρ_o h_o = ρ_w h_w or ρ_o / ρ_w = h_w / h_o

So by measuring h_o and h_w we can find the relative density of the oil (ρ_o/ρ_w), and its density if the density of water is known.

Example 1. The oil column is 10 cm and the water column is 8 cm: relative density of the oil = 8/10 = 0.8; ρ_o = 800 kg/m³.

Example 2 (unequal arms, book-style). A U-tube has a narrow arm of cross-section 1 cm² and a wide arm of 2 cm². Water is in the tube, and 6 cm of oil of density 900 kg/m³ is poured into the narrow arm. Height of the water above the separating level: h_w = ρ_o h_o/ρ_w = 900 × 6/1000 = 5.4 cm. (The cross-sectional areas of the arms do not matter, only heights.)

Example 3 (water and mercury). Mercury (13 600 kg/m³) forms a column of 5 cm in one arm; the water column above the mercury level in the other arm: h_w = 13 600 × 5/1000 = 68 cm.

Atmospheric pressure

• is the pressure produced by the weight of the column of air above a unit area of the Earth's surface.
• Going higher, the column of air above you gets shorter, so the atmospheric pressure decreases with height.

• Torricelli took a glass tube about 1 m long, closed at one end, completely filled it with mercury, and turned it upside down with its open end under the surface of mercury in a dish.
• The mercury in the tube dropped until the vertical height of its surface above the surface of mercury in the dish was about 0.76 m.

• The space above the mercury is a vacuum (the Torricelli vacuum), apart from a little mercury vapour whose pressure is negligible.

• The vertical height h does not change if the tube is tilted or if the tube is wider or narrower — only the length of mercury in the tube changes when it is tilted.

• Take two points A (outside the tube, at the mercury surface in the dish) and B (inside the tube, at the same horizontal level). The pressure at B = the pressure at A = atmospheric pressure. The pressure at B is produced by the mercury column above it. So

  P_a = ρ_Hg g h

Standard atmospheric pressure is the pressure that supports a column of mercury 0.76 m high at 0 °C (cross-section 1 m²) at sea level; it is the pressure at STP (standard temperature and pressure: 0.76 m Hg and 0 °C), a term often used with gas laws.

Using ρ_Hg = 13 595 kg/m³ at 0 °C and g = 9.8 m/s²:

P_a = 1 atm = 0.76 × 13 595 × 9.8 = 1.013 × 10⁵ N/m²

• The height of the mercury column does not depend on the cross-section of the tube, nor on tilting it, nor on the amount of mercury in the dish.
• It depends on the atmospheric pressure, the density of the liquid used and g.
• (A water barometer would need a column 13.6 times taller: about 10.3 m.)

Example 1. The barometer height on a hilltop is 0.72 m Hg. P_a = 0.72 × 13 600 × 9.8 = 9.60 × 10⁴ N/m².

Example 2. Water barometer height: P_a = ρ_w g h → h = 1.013 × 10⁵/(1000 × 9.8) = 10.3 m.

Example 3. Convert 1.5 atm: in cm Hg = 1.5 × 76 = 114 cm Hg; in Pa = 1.5 × 1.013 × 10⁵ = 1.52 × 10⁵ Pa.

Example 4 (change of height). A mercury barometer reads 76 cm Hg at the bottom of a tower and 74 cm Hg at the top. The pressure difference = 2 cm Hg = 0.02 × 13 600 × 9.8 = 2666 N/m². If the air density is about 1.2 kg/m³ then ρ_air g H = 2666 → H = 2666/(1.2 × 9.8) ≈ 227 m.

• The pressure on the two sides of the eardrum is normally equal.
• When we rise quickly (in a lift, aeroplane or climbing a mountain), the outside pressure decreases, and the higher pressure inside the middle ear pushes the eardrum outward, producing a feeling of tension.
• Swallowing or chewing opens the Eustachian tube and equalises the pressure.

Why don't we feel the huge atmospheric pressure (more than 10 tonnes per square metre)? Because the pressure of the fluids inside our bodies equals the outside pressure, so the forces on the surfaces of the body are balanced.

• A manometer measures the difference between the pressure of a gas in a container and the atmospheric pressure.
• It is a U-shaped tube containing a liquid of known density (usually mercury, or water/oil for small pressures).
• One arm is connected to the gas container; the other is open to the air.

• Gas pressure greater than atmospheric:

  • the liquid is lower in the arm connected to the gas.
  • Using two points at the same level in the same liquid:

  P = P_a + ρ g h

• Gas pressure less than atmospheric: the liquid in the open arm is lower:

  P = P_a − ρ g h

The pressure difference (gauge pressure) is ΔP = P − P_a = ρ g h (positive or negative).

Choice of manometer liquid.

• Mercury for large pressures (dense, so the height difference is small); a light liquid (water, oil) for small pressure differences, which gives a large and readable height difference.
• The liquid must not react with or dissolve the gas and should have a low vapour pressure.

Example 1. A mercury manometer connected to a gas container shows the mercury in the open arm 24 cm higher than in the arm connected to the container. The gas pressure P = 76 + 24 = 100 cm Hg = 100/76 = 1.316 atm = 1.316 × 1.013 × 10⁵ = 1.333 × 10⁵ N/m².

Example 2. If the mercury in the open arm were 18 cm lower, then P = 76 − 18 = 58 cm Hg (below atmospheric: a partial vacuum).

Example 3 (water manometer). A water manometer shows a height difference of 40 cm. ΔP = ρgh = 1000 × 9.8 × 0.4 = 3920 N/m².

Example 4 (mercury, SI). In a manometer h = 25 cm of mercury, ρ = 13 600, g = 9.8: ΔP = 13 600 × 9.8 × 0.25 = 3.332 × 10⁴ N/m²; the absolute gas pressure = 1.013 × 10⁵ + 0.333 × 10⁵ = 1.346 × 10⁵ N/m².

• 1.
• Blood pressure. Blood is a viscous fluid pumped by the heart through arteries and veins.
• Blood flow is normally steady.
• If it becomes turbulent it produces noise, detected by a stethoscope on the artery while measuring blood pressure.
• Two values are recorded:

• Systolic pressure: the maximum pressure in the artery when the heart muscle contracts and pushes blood from the left ventricle into the aorta. Normal: about 120 torr.
• Diastolic pressure: the minimum pressure when the heart muscle relaxes. Normal: about 80 torr. (The cuff is placed on the upper arm at heart level so that the height of the blood column does not add or subtract ρgh to the reading. Long sitting increases the risk of leg blood clots because blood flow in the legs is slow; moving and exercising the legs helps.)

• 2.
• Car-tyre pressure. A tyre filled with air at high pressure has a small contact area with the road.
• A tyre with too low a pressure has a large contact area, so more friction and more heat, and wear.
• Pressure is measured by a tyre-pressure gauge (a piston pushed against a spring with a graduated scale).

Example 1 (tyre). A tyre gauge shows 2.5 × 10⁵ N/m² above the atmosphere. Absolute pressure = 2.5 × 10⁵ + 1.013 × 10⁵ = 3.51 × 10⁵ N/m² (3.5 atm).

Example 2. Blood pressure of 120 torr = 120 mm Hg = 0.12 × 13 600 × 9.8 = 1.6 × 10⁴ N/m².

Consider a liquid in a container fitted with a piston. At a point A at depth h below the piston the pressure is P = P₁ + ρgh, where P₁ is the pressure just under the piston. If we add a weight on the piston, the pressure at the piston surface rises by ΔP. The piston does not move because the liquid is incompressible, but the pressure at every point increases by the same ΔP.

Pascal's principle (law): When pressure is applied to a liquid enclosed in a container, the pressure is transmitted in full to every part of the liquid and to the walls of the container.

(If the pressure is increased too much, the container may burst.)

• It consists of two cylinders connected by a pipe and filled with a suitable liquid (oil).
• One cylinder has a small piston of area a; the other has a large piston of area A.

A force f on the small piston creates a pressure P = f/a. By Pascal's principle this pressure is transmitted to the large piston, which is at the same level, so P = F/A also.

f / a = F / A → F = f · (A / a)

Hence a small force on the small piston produces a large force on the large piston.

Mechanical advantage of the press:

η = F / f = A / a (a ratio, no unit)

Energy conservation. If the small piston moves down a distance y₁ and the large piston moves up y₂, then (ideal press, efficiency 100 %) the work done on the small piston equals the work done by the large piston:

f y₁ = F y₂ → F/f = y₁/y₂ = A/a

So η = F/f = A/a = y₁/y₂. The large piston moves a smaller distance than the small one — we gain force but not work.

Because the area depends on the square of the radius or diameter, if the ratio of the radii is r_large : r_small = n, then η = n². (E.g. a radius ratio of 9 : 1 between the large and the small piston gives η = 81.)

1. Hydraulic brakes in cars: the pedal pushes a piston in the master cylinder; the pressure is transmitted through the brake fluid to the pistons of the wheel cylinders, which push brake shoes against the drum (rear drum brakes) or pads against the disc (front disc brakes). A small foot force produces large friction forces at all four wheels, and the pressure is the same to all wheels.
2. Hydraulic lift (car lift in a garage).
3. Hydraulic excavator and other hydraulic machines (jacks, presses, aircraft control systems).

Example 1. In a hydraulic press the small piston has diameter 4 cm and the large piston has diameter 40 cm. A force of 50 N is applied to the small piston. (g = 10 m/s²)

• Area ratio A/a = (40/4)² = 100, so η = 100.
• F = 50 × 100 = 5000 N; the largest mass it can lift = 5000/10 = 500 kg.
• To raise the large piston by 2 cm: y₁ = η y₂ = 100 × 2 = 200 cm.

Example 2 (car lift). A car of mass 1200 kg (weight 12 000 N) rests on a piston of radius 20 cm; the small piston has a radius of 2 cm. f = F(a/A) = 12 000 × (2/20)² = 12 000/100 = 120 N.

Example 3 (pressure). What pressure is transmitted if 120 N acts on a small piston of area 4 × 10⁻⁴ m²? P = 120/4 × 10⁻⁴ = 3 × 10⁵ N/m².

Example 4 (efficiency idea). A press has η = 40. A force of 250 N acts on the small piston, which is pushed 10 cm. F = 10 000 N; the large piston rises y₂ = 10/40 = 0.25 cm. Work input = 250 × 0.10 = 25 J; work output = 10 000 × 0.0025 = 25 J (equal).

Example 5 (given η and F). A hydraulic press lifts a 2000 N load on the large piston of area 0.05 m² with a small-piston area of 2.5 × 10⁻³ m². η = 0.05/2.5 × 10⁻³ = 20, so f = 2000/20 = 100 N.

• Which factor does not affect the pressure at the bottom of a liquid-filled container? The area of the base (and the shape of the vessel): pressure depends on depth, density, g and atmospheric pressure.
• Which factor does not affect the height of mercury in a barometer? The cross-sectional area of the tube.
• The ratio of the radii of the pistons of a hydraulic press is 2 : 9; what is the ratio between the forces on the two pistons? f/F = a/A = (2/9)² = 4/81.
• Prove P = P_a + ρgh: see Lesson 2.2 above.

• P = F/A (pascal = N/m²).
• In a liquid: P = P_a + ρgh; same level, same pressure; connecting vessels; U-tube of immiscible liquids ρ₁h₁ = ρ₂h₂.
• Atmospheric pressure: P_a = ρ_Hg g h; 1 atm = 1.013 × 10⁵ Pa = 760 mm Hg = 1.013 bar; STP.
• Manometer: ΔP = P − P_a = ρgh.
• Blood pressure: 120/80 torr; tyre gauge.
• Pascal's principle: pressure on an enclosed liquid is transmitted in full; hydraulic press f/a = F/A; η = F/f = A/a = y₁/y₂; applications: brakes, lifts, excavators.