Pressure and Fluid Statics

Thrust is a force acting perpendicular to a surface. Pressure measures thrust per unit area.

For uniform pressure, p = F/A. The same formula gives average pressure when the pressure varies across the area.

Here, F is perpendicular force in newtons and A is area in square metres. The SI unit of pressure is the pascal (Pa).

1 Pa = 1 N/m².

Suppose a 200 N force acts evenly over 0.02 m². The pressure is 200 ÷ 0.02 = 10,000 Pa.

Double the area while keeping the force unchanged. Pressure falls to 5,000 Pa.

A wide base spreads weight over more ground. A sharp blade concentrates its force over a smaller contact area.

Is pressure a vector?

Pressure is a scalar: it has magnitude but no direction. The force produced by pressure on a surface does have a direction.

In a fluid at rest, that force acts perpendicular to the surface. At one point, pressure is the same for every surface orientation.

Equal pressure in different directions does not mean equal pressure at different depths.

A deeper layer supports more fluid above it. For a stationary liquid with constant density, p = p₀ + ρgh.

• p is pressure at depth h, measured in pascals.
• p₀ is pressure at the liquid's surface, measured in pascals.
• ρ, pronounced “rho”, is density in kilograms per cubic metre.
• g is gravitational acceleration in metres per second squared.
• h is vertical depth below the surface, measured in metres.

The extra pressure from the liquid column is ρgh. This expression assumes approximately constant density and gravitational acceleration.

Take water density as 1,000 kg/m³ and g as 10 m/s². At 2 m depth, the extra pressure is 1,000 × 10 × 2 = 20,000 Pa.

Add the surface pressure to find total pressure. If p₀ is 100,000 Pa, total pressure is 120,000 Pa.

Does a wider container produce greater pressure?

At equal depth in the same connected liquid at rest, pressure is equal. Different container shapes do not change this result.

For separate containers, also keep surface pressure, liquid density and g unchanged. Base pressure then depends on liquid depth, rather than base area.

Force and pressure are different. A larger horizontal base experiences a larger pressure force when pressure stays unchanged, because F = pA.

For a net force across a surface, use the pressure difference between its two sides. Equal pressures on opposite sides can cancel.

Pressure differences across the eardrum also explain discomfort during a descent underwater. The pressure imbalance matters, rather than depth alone.

Absolute pressure is measured relative to a vacuum. Gauge pressure is measured relative to the surrounding atmospheric pressure.

pabsolute = patmosphere + pgauge.

An open water surface has zero gauge pressure. Its absolute pressure is still atmospheric pressure.

A gauge pressure below zero means pressure is below the surrounding atmospheric pressure. It does not mean negative absolute pressure in these ordinary gas-pressure examples.

How does a barometer measure the atmosphere?

A barometer measures atmospheric pressure. A mercury barometer balances atmospheric pressure against the weight of a mercury column.

The space above the column has very low pressure. Neglecting mercury vapour pressure gives patmosphere = ρmercury gh.

One standard atmosphere is approximately 1.013 × 10⁵ Pa, equivalent to about 76 cm of mercury. Actual atmospheric pressure varies with altitude and weather.

Atmospheric pressure generally decreases as altitude increases. The 76 cm value is a reference, rather than every sea-level reading.

A manometer measures a pressure difference using a liquid column. In a simple U-tube, the difference follows Δp = ρgΔh when gas-column weights are negligible.

Here, Δh is the vertical difference between the liquid levels. Use the manometer liquid's density in this formula.

Pascal's principle concerns a pressure change applied to an enclosed fluid. The added pressure is transmitted undiminished throughout the fluid and to its container walls.

Depth-related pressure differences can still exist. The principle does not make absolute pressure identical at every height.

For ideal pistons at the same level, F₁/A₁ = F₂/A₂. The forces are the added forces producing the shared pressure increase.

Suppose the input piston has area 2 cm² and the output piston has area 20 cm². The area ratio is ten.

An input force of 50 N produces an ideal output force of 500 N. Equal pressure acts over the larger output area.

Does a hydraulic lift create energy?

No. For an incompressible liquid without leaks, both pistons displace equal liquid volumes.

A₁d₁ = A₂d₂, where d₁ and d₂ are piston travel distances.

In the ten-to-one example, the smaller piston moves ten times farther. The greater output force is balanced by the shorter output travel.

Without losses, input work equals output work. Real systems also experience losses, including friction.

Hydraulic brakes and lifts use pressure transmission. The ideal relations explain their principle without describing every real component.

Archimedes' principle: buoyant force equals the weight of displaced fluid. The principle applies to both partial and complete immersion.

For a uniform fluid, FB = ρfluid g Vdisplaced.

Here, FB is buoyant force in newtons. Vdisplaced is the submerged volume excluding the fluid, measured in cubic metres.

Use the fluid's density, rather than the object's density. For partial immersion, use only the submerged volume.

Consider a block surrounded by stationary water. Pressure on its lower face exceeds pressure on its upper face. The difference produces an upward resultant force.

An object displaces 0.002 m³ of water with density 1,000 kg/m³. Using g = 10 m/s² gives FB = 20 N.

Why does a spring balance read less underwater?

For an object suspended at rest, T + FB = mg. Here, T is the upward support force measured by the spring balance.

Therefore, apparent weight = T = mg − FB. Assume the object touches neither the bottom nor the container walls.

An object with weight 30 N receives 20 N of buoyant force. The balance reads 10 N.

Its mass has not decreased. Water provides part of the upward support previously provided by the balance.

Air also provides buoyancy, but basic calculations often neglect its smaller contribution.

A freely floating object at rest displaces fluid equal to its own weight. Buoyant force and weight balance.

Compare the object's average density with the fluid's density. Average density means total mass divided by the object's overall volume.

For an ordinary rigid object, neglect surface tension and external supports:

A hollow steel ship includes a large enclosed air space. Its overall average density can be below water's density.

For a uniform floating block, Vsubmerged/Vtotal = ρobject/ρfluid. This follows by equating buoyant force with weight.

A block with average density 600 kg/m³ floats in water of density 1,000 kg/m³. The submerged fraction is 600/1,000 = 0.60, or 60%.

Is buoyancy greater in seawater?

Seawater is generally denser than fresh water. The same submerged volume therefore receives greater buoyant force in seawater.

A freely floating object settles with less volume submerged in the denser water. Its final buoyant force still equals its unchanged weight.

This extra support for a given submerged volume helps explain why floating can feel easier in seawater.

Floating and resisting a small tilt are different questions. Three points help explain a floating body's initial stability.

• Centre of gravity, G: the point through which the body's weight acts.
• Centre of buoyancy, B: the centre of the displaced fluid volume.
• Metacentre, M: the intersection used to test stability after a small tilt.

When a floating body tilts slightly, the submerged shape changes. The centre of buoyancy shifts with that shape.

The new vertical buoyancy line intersects the body's tilted centreline at M. If M lies above G, the forces tend to restore the original position.

If M lies below G, the forces tend to increase the small tilt. This is a small-angle stability test; it does not guarantee stability at large tilts.

A liquid boils when its saturated vapour pressure reaches the external pressure.

Saturated vapour pressure is the pressure of vapour in equilibrium with its liquid at that temperature.

Heating increases the liquid's saturated vapour pressure. Lower external pressure therefore allows boiling at a lower temperature.

Pure water boils at about 100°C under one standard atmosphere. On a high mountain, lower atmospheric pressure lowers the boiling temperature in an open pot.

Food simmering in that cooler water may take longer to cook. The lower boiling point comes from reduced pressure, rather than cold mountain air alone.

A pressure cooker raises the pressure above the water. Water can then boil above 100°C, allowing a higher cooking temperature.

Pure liquid water has its maximum density near 4°C at ordinary atmospheric pressure. Cooling water from room temperature towards 4°C reduces its volume.

Further cooling from 4°C towards freezing increases its volume. The same mass then occupies more space, so density falls.

This unusual behaviour helps explain why freshwater lakes usually freeze at the surface first. Colder water below 4°C is less dense than the deeper 4°C water.

Ice is also less dense than liquid water, so it floats. Do not apply the exact 4°C statement unchanged to salt water.